ContentsArticles

Poincaré conjecture 1P versus NP problem 7Hodge conjecture 17Riemann hypothesis 21Navier–Stokes existence and smoothness 44Birch and Swinnerton-Dyer conjecture 48Yang–Mills existence and mass gap 51

ReferencesArticle Sources and Contributors 53Image Sources, Licenses and Contributors 54

Article LicensesLicense 55

Poincaré conjecture 1

Poincaré conjecture

Millennium Prize Problems

P versus NP problem

Hodge conjecture

Poincaré conjecture (solution)

Riemann hypothesis

Yang–Mills existence and mass gap

Navier–Stokes existence and smoothness

Birch and Swinnerton-Dyer conjecture

For compact 2-dimensional surfaces without boundary, if every loop can be continuouslytightened to a point, then the surface is topologically homeomorphic to a 2-sphere

(usually just called a sphere). The Poincaré conjecture asserts that the same is true for3-dimensional surfaces.

In mathematics, the Poincaréconjecture ([pwɛ̃kaʁe],[1]

English: /pwɛn.kɑˈreɪ/ pwen-kar-ay) is atheorem about the characterization ofthe three-dimensional sphere(3-sphere), which is the hyperspherethat bounds the unit ball infour-dimensional space. Theconjecture states:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.An equivalent form of the conjecture involves a coarser form of equivalence than homeomorphism called homotopyequivalence: if a 3-manifold is homotopy equivalent to the 3-sphere, then it is necessarily homeomorphic to it.

Originally conjectured by Henri Poincaré, the theorem concerns a space that locally looks like ordinarythree-dimensional space but is connected, finite in size, and lacks any boundary (a closed 3-manifold). The Poincaréconjecture claims that if such a space has the additional property that each loop in the space can be continuouslytightened to a point, then it is necessarily a three-dimensional sphere. An analogous result has been known in higherdimensions for some time.After nearly a century of effort by mathematicians, Grigori Perelman presented a proof of the conjecture in threepapers made available in 2002 and 2003 on arXiv. The proof followed the program of Richard Hamilton. Severalhigh-profile teams of mathematicians have verified that Perelman's proof is correct.The Poincaré conjecture, before being proven, was one of the most important open questions in topology. It is one ofthe seven Millennium Prize Problems, for which the Clay Mathematics Institute offered a $1,000,000 prize for thefirst correct solution. Perelman's work survived review and was confirmed in 2006, leading to his being offered aFields Medal, which he declined. Perelman was awarded the Millennium Prize on March 18, 2010.[2] On July 1,2010, he turned down the prize saying that he believes his contribution in proving the Poincaré conjecture was nogreater than that of U.S. mathematician Richard Hamilton (who first suggested a program for the solution).[3] [4] ThePoincaré conjecture is the first and, as of April 2011, the only solved Millennium problem.On December 22, 2006, the journal Science honored Perelman's proof of the Poincaré conjecture as the scientific"Breakthrough of the Year", the first time this had been bestowed in the area of mathematics.[5]

Poincaré conjecture 2

History

Poincaré's questionAt the beginning of the 20th century, Henri Poincaré was working on the foundations of topology—what would laterbe called combinatorial topology and then algebraic topology. He was particularly interested in what topologicalproperties characterized a sphere.Poincaré claimed in 1900 that homology, a tool he had devised based on prior work by Enrico Betti, was sufficient totell if a 3-manifold was a 3-sphere. However, in a 1904 paper he described a counterexample to this claim, a spacenow called the Poincaré homology sphere. The Poincaré sphere was the first example of a homology sphere, amanifold that had the same homology as a sphere, of which many others have since been constructed. To establishthat the Poincaré sphere was different from the 3-sphere, Poincaré introduced a new topological invariant, thefundamental group, and showed that the Poincaré sphere had a fundamental group of order 120, while the 3-spherehad a trivial fundamental group. In this way he was able to conclude that these two spaces were, indeed, different.In the same paper, Poincaré wondered whether a 3-manifold with the homology of a 3-sphere and also trivialfundamental group had to be a 3-sphere. Poincaré's new condition—i.e., "trivial fundamental group"—can berestated as "every loop can be shrunk to a point."The original phrasing was as follows:

Consider a compact 3-dimensional manifold V without boundary. Is it possible that the fundamental group ofV could be trivial, even though V is not homeomorphic to the 3-dimensional sphere?

Poincaré never declared whether he believed this additional condition would characterize the 3-sphere, butnonetheless, the statement that it does is known as the Poincaré conjecture. Here is the standard form of theconjecture:

Every simply connected, closed 3-manifold is homeomorphic to the 3-sphere.

Attempted solutionsThis problem seems to have lain dormant for a time, until J. H. C. Whitehead revived interest in the conjecture, whenin the 1930s he first claimed a proof, and then retracted it. In the process, he discovered some interesting examples ofsimply connected non-compact 3-manifolds not homeomorphic to R3, the prototype of which is now called theWhitehead manifold.In the 1950s and 1960s, other mathematicians were to claim proofs only to discover a flaw. Influentialmathematicians such as Bing, Haken, Moise, and Papakyriakopoulos attacked the conjecture. In 1958 Bing proved aweak version of the Poincaré conjecture: if every simple closed curve of a compact 3-manifold is contained in a3-ball, then the manifold is homeomorphic to the 3-sphere.[6] Bing also described some of the pitfalls in trying toprove the Poincaré conjecture.[7]

Over time, the conjecture gained the reputation of being particularly tricky to tackle. John Milnor commented thatsometimes the errors in false proofs can be "rather subtle and difficult to detect."[8] Work on the conjecture improvedunderstanding of 3-manifolds. Experts in the field were often reluctant to announce proofs, and tended to view anysuch announcement with skepticism. The 1980s and 1990s witnessed some well-publicized fallacious proofs (whichwere not actually published in peer-reviewed form).[9] [10]

An exposition of attempts to prove this conjecture can be found in the non-technical book Poincaré's Prize byGeorge Szpiro.[11]

Poincaré conjecture 3

DimensionsThe classification of closed surfaces gives an affirmative answer to the analogous question in two dimensions. Fordimensions greater than three, one can pose the Generalized Poincaré conjecture: is a homotopy n-spherehomeomorphic to the n-sphere? A stronger assumption is necessary; in dimensions four and higher there aresimply-connected manifolds which are not homeomorphic to an n-sphere.Historically, while the conjecture in dimension three seemed plausible, the generalized conjecture was thought to befalse. In 1961 Stephen Smale shocked mathematicians by proving the Generalized Poincaré conjecture fordimensions greater than four and extended his techniques to prove the fundamental h-cobordism theorem. In 1982Michael Freedman proved the Poincaré conjecture in dimension four. Freedman's work left open the possibility thatthere is a smooth four-manifold homeomorphic to the four-sphere which is not diffeomorphic to the four-sphere.This so-called smooth Poincaré conjecture, in dimension four, remains open and is thought to be very difficult.Milnor's exotic spheres show that the smooth Poincaré conjecture is false in dimension seven, for example.These earlier successes in higher dimensions left the case of three dimensions in limbo. The Poincaré conjecture wasessentially true in both dimension four and all higher dimensions for substantially different reasons. In dimensionthree, the conjecture had an uncertain reputation until the geometrization conjecture put it into a frameworkgoverning all 3-manifolds. John Morgan wrote:[12]

It is my view that before Thurston's work on hyperbolic 3-manifolds and . . . the Geometrization conjecturethere was no consensus among the experts as to whether the Poincaré conjecture was true or false. AfterThurston's work, notwithstanding the fact that it had no direct bearing on the Poincaré conjecture, a consensusdeveloped that the Poincaré conjecture (and the Geometrization conjecture) were true.

Hamilton's program and Perelman's solution

Several stages of the Ricci flow on atwo-dimensional manifold.

Hamilton's program was started in his 1982 paper in which he introduced theRicci flow on a manifold and showed how to use it to prove some specialcases of the Poincaré conjecture.[13] In the following years he extended thiswork, but was unable to prove the conjecture. The actual solution was notfound until Grigori Perelman published his papers using ideas fromHamilton's work.

In late 2002 and 2003 Perelman posted three papers on the arXiv.[14] [15] [16]

In these papers he sketched a proof of the Poincaré conjecture and a moregeneral conjecture, Thurston's geometrization conjecture, completing theRicci flow program outlined earlier by Richard Hamilton.

From May to July 2006, several groups presented papers that filled in thedetails of Perelman's proof of the Poincaré conjecture, as follows:• Bruce Kleiner and John W. Lott posted a paper on the arXiv in May 2006

which filled in the details of Perelman's proof of the geometrizationconjecture.[17]

• Huai-Dong Cao and Xi-Ping Zhu published a paper in the June 2006 issueof the Asian Journal of Mathematics giving a complete proof of thePoincaré and geometrization conjectures.[18] They initially claimed theproof as their own achievement based on the "Hamilton-Perelman theory",but later retracted the original version of their paper, and posted a revised version, in which they referred to theirwork as the more modest "exposition of Hamilton–Perelman's proof".[19] They were also forced to publish anerratum disclosing that they had failed to cite properly the previous work of Kleiner and Lott published in 2003. In the same issue, the AJM editorial board issued an apology for what it called "incautions" in the Cao–Zhu

Poincaré conjecture 4

paper.• John Morgan and Gang Tian posted a paper on the arXiv in July 2006 which gave a detailed proof of just the

Poincaré Conjecture (which is somewhat easier than the full geometrization conjecture)[20] and expanded this to abook.[21]

All three groups found that the gaps in Perelman's papers were minor and could be filled in using his owntechniques.On August 22, 2006, the ICM awarded Perelman the Fields Medal for his work on the conjecture, but Perelmanrefused the medal.[22] [23] [24] John Morgan spoke at the ICM on the Poincaré conjecture on August 24, 2006,declaring that "in 2003, Perelman solved the Poincaré Conjecture."[25]

In December 2006, the journal Science honored the proof of Poincaré conjecture as the Breakthrough of the Year andfeatured it on its cover.[5]

Ricci flow with surgeryHamilton's program for proving the Poincaré conjecture involves first putting a Riemannian metric on the unknownsimply connected closed 3-manifold. The idea is to try to improve this metric; for example, if the metric can beimproved enough so that it has constant curvature, then it must be the 3-sphere. The metric is improved using theRicci flow equations;

where g is the metric and R its Ricci curvature, and one hopes that as the time t increases the manifold becomeseasier to understand. Ricci flow expands the negative curvature part of the manifold and contracts the positivecurvature part.In some cases Hamilton was able to show that this works; for example, if the manifold has positive Ricci curvatureeverywhere he showed that the manifold becomes extinct in finite time under Ricci flow without any othersingularities. (In other words, the manifold collapses to a point in finite time; it is easy to describe the structure justbefore the manifold collapses.) This easily implies the Poincaré conjecture in the case of positive Ricci curvature.However in general the Ricci flow equations lead to singularities of the metric after a finite time. Perelman showedhow to continue past these singularities: very roughly, he cuts the manifold along the singularities, splitting themanifold into several pieces, and then continues with the Ricci flow on each of these pieces. This procedure isknown as Ricci flow with surgery.A special case of Perelman's theorems about Ricci flow with surgery is given as follows.

The Ricci flow with surgery on a closed oriented 3-manifold is well defined for all time. If the fundamentalgroup is a free product of finite groups and cyclic groups then the Ricci flow with surgery becomes extinct infinite time, and at all times all components of the manifold are connected sums of S2 bundles over S1 andquotients of S3.

This result implies the Poincaré conjecture because it is easy to check it for the possible manifolds listed in theconclusion.The condition on the fundamental group turns out to be necessary (and sufficient) for finite time extinction, and inparticular includes the case of trivial fundamental group. It is equivalent to saying that the prime decomposition ofthe manifold has no acyclic components, and turns out to be equivalent to the condition that all geometric pieces ofthe manifold have geometries based on the two Thurston geometries S2×R and S3. By studying the limit of themanifold for large time, Perelman proved Thurston's geometrization conjecture for any fundamental group: at largetimes the manifold has a thick-thin decomposition, whose thick piece has a hyperbolic structure, and whose thinpiece is a graph manifold, but this extra complication is not necessary for proving just the Poincaré conjecture.[26]

Poincaré conjecture 5

Notes[1] "Poincaré, Jules Henri" (http:/ / www. bartleby. com/ 61/ 3/ P0400300. html). The American Heritage Dictionary of the English Language

(fourth edition ed.). Boston: Houghton Mifflin Company. 2000. ISBN 0-395-82517-2. . Retrieved 2007-05-05..[2] Clay Mathematics Institute (March 18, 2010). "Prize for Resolution of the Poincaré Conjecture Awarded to Dr. Grigoriy Perelman" (http:/ /

www. claymath. org/ poincare/ millenniumPrizeFull. pdf) (PDF). Press release. . Retrieved March 18, 2010. "The Clay Mathematics Institute(CMI) announces today that Dr. Grigoriy Perelman of St. Petersburg, Russia, is the recipient of the Millennium Prize for resolution of thePoincaré conjecture."

[3] Последнее "нет" доктора Перельмана (http:/ / www. interfax. ru/ society/ txt. asp?id=143603), Interfax 1 July 2010[4] Ritter, Malcolm (1 July 2010). "Russian mathematician rejects million prize" (http:/ / www. boston. com/ news/ science/ articles/ 2010/ 07/

01/ russian_mathematician_rejects_1_million_prize/ ?p1=Well_MostPop_Emailed1). The Boston Globe. .[5] Mackenzie, Dana (2006-12-22). "The Poincaré Conjecture--Proved" (http:/ / www. sciencemag. org/ cgi/ content/ full/ 314/ 5807/ 1848).

Science (American Association for the Advancement of Science) 314 (5807): 1848–1849. doi:10.1126/science.314.5807.1848.PMID 17185565. ISSN: 0036-8075. .

[6] Bing, RH (1958). "Necessary and sufficient conditions that a 3-manifold be S3". The Annals of Mathematics, 2nd Ser. 68 (1): 17–37.doi:10.2307/1970041. JSTOR 1970041.

[7] Bing, RH (1964). "Some aspects of the topology of 3-manifolds related to the Poincaré conjecture". Lectures on Modern Mathematics, Vol.II. New York: Wiley. pp. 93–128.

[8] Milnor, John (2004). "The Poincaré Conjecture 99 Years Later: A Progress Report" (http:/ / www. math. sunysb. edu/ ~jack/ PREPRINTS/poiproof. pdf) (PDF). . Retrieved 2007-05-05.

[9] Taubes, Gary (July 1987). "What happens when hubris meets nemesis". Discover 8: 66–77.[10] Matthews, Robert (9 April 2002). "$1 million mathematical mystery "solved"" (http:/ / www. newscientist. com/ article. ns?id=dn2143).

NewScientist.com. . Retrieved 2007-05-05.[11] Szpiro, George (July 29, 2008). Poincaré's Prize: The Hundred-Year Quest to Solve One of Math's Greatest Puzzles. Plume.

ISBN 978-0-452-28964-2.[12] Morgan, John W., Recent progress on the Poincaré conjecture and the classification of 3-manifolds. Bull. Amer. Math. Soc. (N.S.) 42

(2005), no. 1, 57–78[13] Hamilton, Richard (1982). "Three-manifolds with positive Ricci curvature". Journal of Differential Geometry 17: 255–306. Reprinted in:

Cao, H.D.; et al. (Editors) (2003). Collected Papers on Ricci Flow. International Press. ISBN 978-1571461100.[14] Perelman, Grigori (2002). The entropy formula for the Ricci flow and its geometric applications. arXiv:math.DG/0211159.[15] Perelman, Grigori (2003). Ricci flow with surgery on three-manifolds. arXiv:math.DG/0303109.[16] Perelman, Grigori (2003). Finite extinction time for the solutions to the Ricci flow on certain three-manifolds. arXiv:math.DG/0307245.[17] Kleiner, Bruce; John W. Lott (2006). Notes on Perelman's Papers. arXiv:math.DG/0605667.[18] Cao, Huai-Dong; Xi-Ping Zhu (June 2006). "A Complete Proof of the Poincaré and Geometrization Conjectures – application of the

Hamilton-Perelman theory of the Ricci flow" (http:/ / www. intlpress. com/ AJM/ p/ 2006/ 10_2/ AJM-10-2-165-492. pdf) (PDF). AsianJournal of Mathematics 10 (2). .

[19] Cao, Huai-Dong and Zhu, Xi-Ping (December 3, 2006). "Hamilton–Perelman's Proof of the Poincaré Conjecture and the GeometrizationConjecture". arXiv.org. arXiv:math.DG/0612069.

[20] Morgan, John; Gang Tian (2006). Ricci Flow and the Poincaré Conjecture. arXiv:math.DG/0607607.[21] Morgan, John; Gang Tian (2007). Ricci Flow and the Poincaré Conjecture. Clay Mathematics Institute. ISBN 0821843281.[22] Nasar, Sylvia; David Gruber (August 28, 2006). "Manifold destiny". The New Yorker: pp. 44–57. On-line version at the New Yorker website

(http:/ / www. newyorker. com/ fact/ content/ articles/ 060828fa_fact2).[23] Chang, Kenneth (August 22, 2006). "Highest Honor in Mathematics Is Refused" (http:/ / www. nytimes. com/ 2006/ 08/ 22/ science/

22cnd-math. html?hp& ex=1156305600& en=aa3a9d418768062c& ei=5094& partner=homepage). New York Times. .[24] "Reclusive Russian solves 100-year-old maths problem" (http:/ / www. chinadaily. com. cn/ cndy/ 2006-08/ 23/ content_671442. htm).

China Daily: p. 7. 23 August 2006. .[25] A Report on the Poincaré Conjecture. Special lecture by John Morgan.[26] Terence Tao wrote an exposition of Ricci flow with surgery in: Tao, Terence (2006). Perelman's proof of the Poincaré conjecture: a

nonlinear PDE perspective. arXiv:math.DG/0610903.

Poincaré conjecture 6

External links• The Poincaré conjecture described (http:/ / www. claymath. org/ prizeproblems/ poincare. htm) by the Clay

Mathematics Institute.• The Poincaré Conjecture (video) (http:/ / www. youtube. com/ watch?v=AUoaTrQTM5o) Brief visual overview

of the Poincaré Conjecture, background and solution.• The Geometry of 3-Manifolds(video) (http:/ / athome. harvard. edu/ threemanifolds/ ) A public lecture on the

Poincaré and geometrization conjectures, given by C. McMullen at Harvard in 2006.• Bruce Kleiner (Yale) and John W. Lott (University of Michigan): "Notes & commentary on Perelman's Ricci flow

papers" (http:/ / www. math. lsa. umich. edu/ ~lott/ ricciflow/ perelman. html).• Stephen Ornes, What is The Poincaré Conjecture? (http:/ / www. seedmagazine. com/ news/ 2006/ 08/

what_is_the_poincar_conjecture. php), Seed Magazine, 25 August 2006.• The slides (http:/ / www. mcm. ac. cn/ Active/ yau_new. pdf) used by Yau in a popular talk on the Poincaré

conjecture.• "The Poincaré Conjecture" (http:/ / www. bbc. co. uk/ radio4/ history/ inourtime/ inourtime_20061102. shtml) –

BBC Radio 4 programme In Our Time, 2 November 2006. Contributors June Barrow-Green, Lecturer in theHistory of Mathematics at the Open University, Ian Stewart, Professor of Mathematics at the University ofWarwick, Marcus du Sautoy, Professor of Mathematics at the University of Oxford, and presenter Melvyn Bragg.

• "Solving an Old Math Problem Nets Award, Trouble" (http:/ / www. npr. org/ templates/ story/ story.php?storyId=6682439) – NPR segment, December 26, 2006.

• Nasar, Sylvia; and Gruber, David (21 August 2006). "Manifold Destiny: A legendary problem and the battle overwho solved it." (http:/ / www. newyorker. com/ fact/ content/ articles/ 060828fa_fact2). The New Yorker.Retrieved 2006-08-24.

P versus NP problem 7

P versus NP problem

Millennium Prize Problems

P versus NP problem

Hodge conjecture

Poincaré conjecture (solution)

Riemann hypothesis

Yang–Mills existence and mass gap

Navier–Stokes existence and smoothness

Birch and Swinnerton-Dyer conjecture

Diagram of complexity classes provided that P ≠ NP. The existenceof problems outside both P and NP-complete in this case was

established by Ladner's theorem.[1]

The P versus NP problem is a major unsolvedproblem in computer science. Informally, it askswhether every problem whose solution can beefficiently checked by a computer can also beefficiently solved by a computer. It was introduced in1971 by Stephen Cook in his paper "The complexity oftheorem proving procedures"[2] and is considered bymany to be the most important open problem in thefield.[3] It is one of the seven Millennium PrizeProblems selected by the Clay Mathematics Institute tocarry a US$ 1,000,000 prize for the first correctsolution.

In essence, the question P = NP? asks:Suppose that solutions to a problem can be verified quickly. Then, can the solutions themselves also becomputed quickly?

The theoretical notion of quick used here is an algorithm that runs in polynomial time. The general class of questionsfor which some algorithm can provide an answer in polynomial time is called "class P" or just "P".

For some questions, there is no known way to find an answer quickly, but if one is provided with informationshowing what the answer is, it may be possible to verify the answer quickly. The class of questions for which ananswer can be verified in polynomial time is called NP.Consider the subset sum problem, an example of a problem that is easy to verify, but whose answer may be difficultto compute. Given a set of integers, does some nonempty subset of them sum to 0? For instance, does a subset of theset {−2, −3, 15, 14, 7, −10} add up to 0? The answer "yes, because {−2, −3, −10, 15} add up to zero" can be quicklyverified with three additions. However, there is no known algorithm to find such a subset in polynomial time (thereis, however, in exponential time, which consists of 2n-1 tries), and indeed such an algorithm cannot exist if the twocomplexity classes are not the same; hence this problem is in NP (quickly checkable) but not necessarily in P(quickly solvable).An answer to the P = NP question would determine whether problems like the subset-sum problem that can beverified in polynomial time can also be solved in polynomial time. If it turned out that P does not equal NP, it wouldmean that there are problems in NP (such as NP-complete problems) that are harder to compute than to verify: theycould not be solved in polynomial time, but the answer could be verified in polynomial time.

P versus NP problem 8

ContextThe relation between the complexity classes P and NP is studied in computational complexity theory, the part of thetheory of computation dealing with the resources required during computation to solve a given problem. The mostcommon resources are time (how many steps it takes to solve a problem) and space (how much memory it takes tosolve a problem).In such analysis, a model of the computer for which time must be analyzed is required. Typically such modelsassume that the computer is deterministic (given the computer's present state and any inputs, there is only onepossible action that the computer might take) and sequential (it performs actions one after the other).In this theory, the class P consists of all those decision problems (defined below) that can be solved on adeterministic sequential machine in an amount of time that is polynomial in the size of the input; the class NPconsists of all those decision problems whose positive solutions can be verified in polynomial time given the rightinformation, or equivalently, whose solution can be found in polynomial time on a non-deterministic machine.[4]

Clearly, P ⊆ NP. Arguably the biggest open question in theoretical computer science concerns the relationshipbetween those two classes:

Is P equal to NP?In a 2002 poll of 100 researchers, 61 believed the answer to be no, 9 believed the answer is yes, and 22 were unsure;8 believed the question may be independent of the currently accepted axioms and so impossible to prove ordisprove.[5]

NP-complete

Euler diagram for P, NP, NP-complete, and NP-hard set of problems

To attack the P = NP question the conceptof NP-completeness is very useful.Informally the NP-complete problems arethe "toughest" problems in NP in the sensethat they are the ones most likely not to bein P. NP-complete problems are a set ofproblems that any other NP-problem can bereduced to in polynomial time, but retain theability to have their solution verified inpolynomial time. In comparison, NP-hardproblems are those at least as hard asNP-complete problems, meaning allNP-problems can be reduced to them, butnot all NP-hard problems are in NP,meaning not all of them have solutionsverifiable in polynomial time.

For instance, the decision problem version of the travelling salesman problem is NP-complete, so any instance ofany problem in NP can be transformed mechanically into an instance of the traveling salesman problem, inpolynomial time. The traveling salesman problem is one of many such NP-complete problems. If any NP-completeproblem is in P, then it would follow that P = NP. Unfortunately, many important problems have been shown to beNP-complete, and as of 2011 not a single fast algorithm for any of them is known.

Based on the definition alone it's not obvious that NP-complete problems exist. A trivial and contrived NP-complete problem can be formulated as: given a description of a Turing machine M guaranteed to halt in polynomial time, does there exist a polynomial-size input that M will accept?[6] It is in NP because (given an input) it is simple to

P versus NP problem 9

check whether or not M accepts the input by simulating M; it is NP-complete because the verifier for any particularinstance of a problem in NP can be encoded as a polynomial-time machine M that takes the solution to be verified asinput. Then the question of whether the instance is a yes or no instance is determined by whether a valid input exists.The first natural problem proven to be NP-complete was the Boolean satisfiability problem. This result came to beknown as Cook–Levin theorem; its proof that satisfiability is NP-complete contains technical details about Turingmachines as they relate to the definition of NP. However, after this problem was proved to be NP-complete, proof byreduction provided a simpler way to show that many other problems are in this class. Thus, a vast class of seeminglyunrelated problems are all reducible to one another, and are in a sense "the same problem".

Harder problemsAlthough it is unknown whether P = NP, problems outside of P are known. A number of succinct problems(problems that operate not on normal input, but on a computational description of the input) are known to beEXPTIME-complete. Because it can be shown that P EXPTIME, these problems are outside P, and so requiremore than polynomial time. In fact, by the time hierarchy theorem, they cannot be solved in significantly less thanexponential time. Examples include finding a perfect strategy for chess (on an N×N board)[7] and some other boardgames.[8]

The problem of deciding the truth of a statement in Presburger arithmetic requires even more time. Fischer andRabin proved in 1974 that every algorithm that decides the truth of Presburger statements has a runtime of at least

for some constant c. Here, n is the length of the Presburger statement. Hence, the problem is known to needmore than exponential run time. Even more difficult are the undecidable problems, such as the halting problem. Theycannot be completely solved by any algorithm, in the sense that for any particular algorithm there is at least one inputfor which that algorithm will not produce the right answer; it will either produce the wrong answer, finish withoutgiving a conclusive answer, or otherwise run forever without producing any answer at all.

Problems in NP not known to be in P or NP-completeIt was shown by Ladner that if P ≠ NP then there exist problems in NP that are neither in P nor NP-complete.[1] Suchproblems are called NP-intermediate problems. The graph isomorphism problem, the discrete logarithm problem andthe integer factorization problem are examples of problems believed to be NP-intermediate. They are some of thevery few NP problems not known to be in P or to be NP-complete.The graph isomorphism problem is the computational problem of determining whether two finite graphs areisomorphic. An important unsolved problem in complexity theory is whether the graph isomorphism problem is in P,NP-complete, or NP-intermediate. The answer is not known, but it is believed that the problem is at least notNP-complete.[9] If graph isomorphism is NP-complete, the polynomial time hierarchy collapses to its secondlevel.[10] Since it is widely believed that the polynomial hierarchy does not collapse to any finite level, it is believedthat graph isomorphism is not NP-complete. The best algorithm for this problem, due to Laszlo Babai and EugeneLuks has run time 2O(√(n log n)) for graphs with n vertices.The integer factorization problem is the computational problem of determining the prime factorization of a giveninteger. Phrased as a decision problem, it is the problem of deciding whether the input has a factor less than k. Noefficient integer factorization algorithm is known, and this fact forms the basis of several modern cryptographicsystems, such as the RSA algorithm. The integer factorization problem is in NP and in co-NP (and even in UP andco-UP[11] ). If the problem is NP-complete, the polynomial time hierarchy will collapse to its first level (i.e., NP willequal co-NP). The best known algorithm for integer factorization is the general number field sieve, which takesexpected time O(e(64/9)1/3(n.log 2)1/3(log (n.log 2))2/3) to factor an n-bit integer. However, the best known quantumalgorithm for this problem, Shor's algorithm, does run in polynomial time. Unfortunately, this fact doesn't say muchabout where the problem lies with respect to non-quantum complexity classes.

P versus NP problem 10

Does P mean "easy"?

The graph shows time (average of 100 instances in msec using a 933 MHz PentiumIII) vs.problem size for knapsack problems for a state-of-the-art specialized

algorithm. Quadratic fit suggests that empirical algorithmic complexity for instanceswith 50–10,000 variables is O((log n)2).[12]

All of the above discussion has assumedthat P means "easy" and "not in P" means"hard". This assumption, known asCobham's thesis, though a common andreasonably accurate assumption incomplexity theory, is not always true inpractice; the size of constant factors orexponents may have practical importance,or there may be solutions that work forsituations encountered in practice despitehaving poor worst-case performance intheory (this is the case for instance for thesimplex algorithm in linearprogramming). Other solutions violate theTuring machine model on which P andNP are defined by introducing conceptslike randomness and quantum computation.

Because of these factors, even if a problem is shown to be NP-complete, and even if P ≠ NP, there may still beeffective approaches to tackling the problem in practice. There are algorithms for many NP-complete problems, suchas the knapsack problem, the travelling salesman problem and the boolean satisfiability problem, that can solve tooptimality many real-world instances in reasonable time. The empirical average-case complexity (time vs. problemsize) of such algorithms can be surprisingly low.

Reasons to believe P ≠ NPAccording to a poll,[5] many computer scientists believe that P ≠ NP. A key reason for this belief is that afterdecades of studying these problems no one has been able to find a polynomial-time algorithm for any of more than3000 important known NP-complete problems (see List of NP-complete problems). These algorithms were soughtlong before the concept of NP-completeness was even defined (Karp's 21 NP-complete problems, among the firstfound, were all well-known existing problems at the time they were shown to be NP-complete). Furthermore, theresult P = NP would imply many other startling results that are currently believed to be false, such as NP = co-NPand P = PH.It is also intuitively argued that the existence of problems that are hard to solve but for which the solutions are easyto verify matches real-world experience.[13]

If P = NP, then the world would be a profoundly different place than we usually assume it to be. There wouldbe no special value in "creative leaps," no fundamental gap between solving a problem and recognizing thesolution once it's found. Everyone who could appreciate a symphony would be Mozart; everyone who couldfollow a step-by-step argument would be Gauss...— Scott Aaronson, MIT

On the other hand, some researchers believe that there is overconfidence in believing P ≠ NP and that researchersshould explore proofs of P = NP as well. For example, in 2002 these statements were made:[5]

The main argument in favor of P ≠ NP is the total lack of fundamental progress in the area of exhaustive search. This is, in my opinion, a very weak argument. The space of algorithms is very large and we are only at the beginning of its exploration. [. . .] The resolution of Fermat's Last Theorem also shows that very simple

P versus NP problem 11

questions may be settled only by very deep theories.—Moshe Y. Vardi, Rice UniversityBeing attached to a speculation is not a good guide to research planning. One should always try both directionsof every problem. Prejudice has caused famous mathematicians to fail to solve famous problems whosesolution was opposite to their expectations, even though they had developed all the methods required.—Anil Nerode, Cornell University

Consequences of proofOne of the reasons the problem attracts so much attention is the consequences of the answer. A proof that P = NPcould have stunning practical consequences, if the proof leads to efficient methods for solving some of the importantproblems in NP. It is also possible that a proof would not lead directly to efficient methods, perhaps if the proof isnon-constructive, or the size of the bounding polynomial is too big to be efficient in practice. The consequences,both positive and negative, arise since various NP-complete problems are fundamental in many fields.Cryptography, for example, relies on certain problems being difficult. A constructive and efficient solution to anNP-complete problem such as 3-SAT would break most existing cryptosystems including public-key cryptography, afoundation for many modern security applications such as secure economic transactions over the Internet, andsymmetric ciphers such as AES or 3DES, used for the encryption of communications data. These would need to bemodified or replaced by information-theoretically secure solutions.On the other hand, there are enormous positive consequences that would follow from rendering tractable manycurrently mathematically intractable problems. For instance, many problems in operations research are NP-complete,such as some types of integer programming, and the travelling salesman problem, to name two of the most famousexamples. Efficient solutions to these problems would have enormous implications for logistics. Many otherimportant problems, such as some problems in protein structure prediction, are also NP-complete;[14] if theseproblems were efficiently solvable it could spur considerable advances in biology.But such changes may pale in significance compared to the revolution an efficient method for solving NP-completeproblems would cause in mathematics itself. According to Stephen Cook,[15]

...it would transform mathematics by allowing a computer to find a formal proof of any theorem which has aproof of a reasonable length, since formal proofs can easily be recognized in polynomial time. Exampleproblems may well include all of the CMI prize problems.

Research mathematicians spend their careers trying to prove theorems, and some proofs have taken decades or evencenturies to find after problems have been stated—for instance, Fermat's Last Theorem took over three centuries toprove. A method that is guaranteed to find proofs to theorems, should one exist of a "reasonable" size, wouldessentially end this struggle.A proof that showed that P ≠ NP would lack the practical computational benefits of a proof that P = NP, but wouldnevertheless represent a very significant advance in computational complexity theory and provide guidance forfuture research. It would allow one to show in a formal way that many common problems cannot be solvedefficiently, so that the attention of researchers can be focused on partial solutions or solutions to other problems. Dueto widespread belief in P ≠ NP, much of this focusing of research has already taken place.[16]

A "not equal" resolution to the P versus NP problem still leaves open the average-case complexity of hard problems in NP. For example, it is possible that SAT requires exponential time in the worst case, but that almost all randomly selected instances of it are efficiently solvable. Russell Impagliazzo has described five hypothetical "worlds" that could result from different possible resolutions to the average-case complexity question.[17] These range from "Algorithmica", where P=NP and problems like SAT can be solved efficiently in all instances, to "Cryptomania", where P≠NP and generating hard instances of problems outside P is easy, with three intermediate possibilities reflecting different possible distributions of difficulty over instances of NP-hard problems. The "world" where P≠NP

P versus NP problem 12

but all problems in NP are tractable in the average case is called "Heuristica" in the paper. A Princeton Universityworkshop in 2009 studied the status of the five worlds.[18]

Results about difficulty of proofAlthough the P = NP? problem itself remains open, despite a million-dollar prize and a huge amount of dedicatedresearch, efforts to solve the problem have led to several new techniques. In particular, some of the most fruitfulresearch related to the P = NP problem has been in showing that existing proof techniques are not powerful enoughto answer the question, thus suggesting that novel technical approaches are required.As additional evidence for the difficulty of the problem, essentially all known proof techniques in computationalcomplexity theory fall into one of the following classifications, each of which is known to be insufficient to provethat P ≠ NP:

Classification Definition

Relativizingproofs

Imagine a world where every algorithm is allowed to make queries to some fixed subroutine called an oracle, and the running timeof the oracle is not counted against the running time of the algorithm. Most proofs (especially classical ones) apply uniformly in aworld with oracles regardless of what the oracle does. These proofs are called relativizing. In 1975, Baker, Gill, and Solovayshowed that P = NP with respect to some oracles, while P ≠ NP for other oracles.[19] Since relativizing proofs can only provestatements that are uniformly true with respect to all possible oracles, this showed that relativizing techniques cannot resolve P =NP.

Natural proofs In 1993, Alexander Razborov and Steven Rudich defined a general class of proof techniques for circuit complexity lower bounds,called natural proofs. At the time all previously known circuit lower bounds were natural, and circuit complexity was considered avery promising approach for resolving P = NP. However, Razborov and Rudich showed that, if one-way functions exist, then nonatural proof method can distinguish between P and NP. Although one-way functions have never been formally proven to exist,most mathematicians believe that they do, and a proof or disproof of their existence would be a much stronger statement than thequantification of P relative to NP. Thus it is unlikely that natural proofs alone can resolve P = NP.

Algebrizingproofs

After the Baker-Gill-Solovay result, new non-relativizing proof techniques were successfully used to prove that IP = PSPACE.However, in 2008, Scott Aaronson and Avi Wigderson showed that the main technical tool used in the IP = PSPACE proof,known as arithmetization, was also insufficient to resolve P = NP.[20]

These barriers are another reason why NP-complete problems are useful: if a polynomial-time algorithm can bedemonstrated for an NP-complete problem, this would solve the P = NP problem in a way not excluded by the aboveresults.These barriers have also led some computer scientists to suggest that the P versus NP problem may be independentof standard axiom systems like ZFC (cannot be proved or disproved within them). The interpretation of anindependence result could be that either no polynomial-time algorithm exists for any NP-complete problem, andsuch a proof cannot be constructed in (e.g.) ZFC, or that polynomial-time algorithms for NP-complete problems mayexist, but it's impossible to prove in ZFC that such algorithms are correct.[21] However, if it can be shown, usingtechniques of the sort that are currently known to be applicable, that the problem cannot be decided even with muchweaker assumptions extending the Peano axioms (PA) for integer arithmetic, then there would necessarily existnearly-polynomial-time algorithms for every problem in NP.[22] Therefore, if one believes (as most complexitytheorists do) that not all problems in NP have efficient algorithms, it would follow that proofs of independence usingthose techniques cannot be possible. Additionally, this result implies that proving independence from PA or ZFCusing currently known techniques is no easier than proving the existence of efficient algorithms for all problems inNP.

P versus NP problem 13

Claimed solutionsWhile the P versus NP problem is generally considered unsolved,[23] many amateur and some professionalresearchers have claimed solutions. Woeginger (2010) has a comprehensive list.[24] An August 2010 claim of proofthat P ≠ NP, by Vinay Deolalikar, researcher at HP Labs, Palo Alto, received heavy Internet and press attention afterbeing initially described as "seem[ing] to be a relatively serious attempt" by two leading specialists.[25] The proof hasbeen reviewed publicly by academics,[26] [27] and Neil Immerman, an expert in the field, had pointed out twopossibly fatal errors in the proof.[28] As of September 15, 2010, Deolalikar was reported to be working on a detailedexpansion of his attempted proof.[29] However, the general consensus amongst theoretical computer scientists is nowthat the attempted proof is not correct, or even a significant advancement in our understanding of the problem.

Logical characterizationsThe P = NP problem can be restated in terms of expressible certain classes of logical statements, as a result of workin descriptive complexity. All languages (of finite structures with a fixed signature including a linear order relation)in P can be expressed in first-order logic with the addition of a suitable least fixed point combinator (effectively, this,in combination with the order, allows the definition of recursive functions); indeed, (as long as the signature containsat least one predicate or function in addition to the distinguished order relation [so that the amount of space taken tostore such finite structures is actually polynomial in the number of elements in the structure]), this preciselycharacterizes P. Similarly, NP is the set of languages expressible in existential second-order logic—that is,second-order logic restricted to exclude universal quantification over relations, functions, and subsets. The languagesin the polynomial hierarchy, PH, correspond to all of second-order logic. Thus, the question "is P a proper subset ofNP" can be reformulated as "is existential second-order logic able to describe languages (of finite linearly orderedstructures with nontrivial signature) that first-order logic with least fixed point cannot?". The word "existential" caneven be dropped from the previous characterization, since P = NP if and only if P = PH (as the former wouldestablish that NP = co-NP, which in turn implies that NP = PH). PSPACE = NPSPACE as established Savitch'stheorem, this follows directly from the fact that the square of a polynomial function is still a polynomial function.However, it is believed, but not proven, a similar relationship may not exist between the polynomial time complexityclasses, P and NP so the question is still open.

Polynomial-time algorithmsNo algorithm for any NP-complete problem is known to run in polynomial time. However, there are algorithms forNP-complete problems with the property that if P = NP, then the algorithm runs in polynomial time (although withenormous constants, making the algorithm impractical). The following algorithm, due to Levin, is such an example.It correctly accepts the NP-complete language SUBSET-SUM, and runs in polynomial time if and only if P = NP:

// Algorithm that accepts the NP-complete language SUBSET-SUM.

//

// This is a polynomial-time algorithm if and only if P=NP.

//

// "Polynomial-time" means it returns "yes" in polynomial time when

// the answer should be "yes", and runs forever when it is "no".

//

// Input: S = a finite set of integers

// Output: "yes" if any subset of S adds up to 0.

// Runs forever with no output otherwise.

// Note: "Program number P" is the program obtained by

// writing the integer P in binary, then

P versus NP problem 14

// considering that string of bits to be a

// program. Every possible program can be

// generated this way, though most do nothing

// because of syntax errors.

FOR N = 1...infinity

FOR P = 1...N

Run program number P for N steps with input S

IF the program outputs a list of distinct integers

AND the integers are all in S

AND the integers sum to 0

THEN

OUTPUT "yes" and HALT

If, and only if, P = NP, then this is a polynomial-time algorithm accepting an NP-complete language. "Accepting"means it gives "yes" answers in polynomial time, but is allowed to run forever when the answer is "no".This algorithm is enormously impractical, even if P = NP. If the shortest program that can solve SUBSET-SUM inpolynomial time is b bits long, the above algorithm will try at least 2b−1 other programs first.

Formal definitions for P and NPConceptually a decision problem is a problem that takes as input some string w over an alphabet , and outputs"yes" or "no". If there is an algorithm (say a Turing machine, or a computer program with unbounded memory) thatcan produce the correct answer for any input string of length n in at most steps, where k and c areconstants independent of the input string, then we say that the problem can be solved in polynomial time and weplace it in the class P. Formally, P is defined as the set of all languages that can be decided by a deterministicpolynomial-time Turing machine. That is,P = where and a deterministic polynomial-time Turing machine is a deterministic Turing machine M that satisfies the followingtwo conditions:

1. ; and2. there exists such that (where O refers to the big O notation),

where and

NP can be defined similarly using nondeterministic Turing machines (the traditional way). However, a modernapproach to define NP is to use the concept of certificate and verifier. Formally, NP is defined as the set oflanguages over a finite alphabet that have a verifier that runs in polynomial time, where the notion of "verifier" isdefined as follows.Let L be a language over a finite alphabet, .

if, and only if, there exists a binary relation and a positive integer k such that thefollowing two conditions are satisfied:

1. For all , such that and ; and2. the language over is decidable by a Turing machine in polynomial

time.

P versus NP problem 15

A Turing machine that decides is called a verifier for L and a y such that is called a certificate of

membership of x in L.In general, a verifier does not have to be polynomial-time. However, for L to be in NP, there must be a verifier thatruns in polynomial time.

Example

Let and

Clearly, the question of whether a given x is a composite is equivalent to the question of whether x is a member of. It can be shown that by verifying that satisfies

the above definition (if we identify natural numbers with their binary representations).also happens to be in P.[30] [31]

Formal definition for NP-completenessThere are many equivalent ways of describing NP-completeness.Let be a language over a finite alphabet .

is NP-complete if, and only if, the following two conditions are satisfied:1. ; and2. any is polynomial-time-reducible to (written as ), where if, and only if, the

following two conditions are satisfied:1. There exists such that ; and2. there exists a polynomial-time Turing machine that halts with on its tape on any input .

Notes[1] R. E. Ladner "On the structure of polynomial time reducibility," J.ACM, 22, pp. 151–171, 1975. Corollary 1.1. ACM site (http:/ / portal. acm.

org/ citation. cfm?id=321877& dl=ACM& coll=& CFID=15151515& CFTOKEN=6184618).[2] Cook, Stephen (1971). "The complexity of theorem proving procedures" (http:/ / portal. acm. org/ citation. cfm?coll=GUIDE& dl=GUIDE&

id=805047). Proceedings of the Third Annual ACM Symposium on Theory of Computing. pp. 151–158. .[3] Lance Fortnow, The status of the P versus NP problem (http:/ / www. cs. uchicago. edu/ ~fortnow/ papers/ pnp-cacm. pdf), Communications

of the ACM 52 (2009), no. 9, pp. 78–86. doi:10.1145/1562164.1562186[4] Sipser, Michael: Introduction to the Theory of Computation, Second Edition, International Edition, page 270. Thomson Course Technology,

2006. Definition 7.19 and Theorem 7.20.[5] William I. Gasarch (June 2002). "The P=?NP poll." (http:/ / www. cs. umd. edu/ ~gasarch/ papers/ poll. pdf) (PDF). SIGACT News 33 (2):

34–47. doi:10.1145/1052796.1052804. . Retrieved 2008-12-29.[6] Scott Aaronson. "PHYS771 Lecture 6: P, NP, and Friends" (http:/ / www. scottaaronson. com/ democritus/ lec6. html). . Retrieved

2007-08-27.[7] Aviezri Fraenkel and D. Lichtenstein (1981). "Computing a perfect strategy for n×n chess requires time exponential in n". J. Comb. Th. A

(31): 199–214.[8] David Eppstein. "Computational Complexity of Games and Puzzles" (http:/ / www. ics. uci. edu/ ~eppstein/ cgt/ hard. html). .[9] Arvind, Vikraman; Kurur, Piyush P. (2006). "Graph isomorphism is in SPP". Information and Computation 204 (5): 835–852.

doi:10.1016/j.ic.2006.02.002.[10] Uwe Schöning, "Graph isomorphism is in the low hierarchy", Proceedings of the 4th Annual Symposium on Theoretical Aspects of

Computer Science, 1987, 114–124; also: Journal of Computer and System Sciences, vol. 37 (1988), 312–323[11] Lance Fortnow. Computational Complexity Blog: Complexity Class of the Week: Factoring. September 13, 2002. http:/ / weblog. fortnow.

com/ 2002/ 09/ complexity-class-of-week-factoring. html[12] Pisinger, D. 2003. "Where are the hard knapsack problems?" Technical Report 2003/08, Department of Computer Science, University of

Copenhagen, Copenhagen, Denmark[13] Scott Aaronson. "Reasons to believe" (http:/ / scottaaronson. com/ blog/ ?p=122). ., point 9.[14] Berger B, Leighton T (1998). "Protein folding in the hydrophobic-hydrophilic (HP) model is NP-complete". J. Comput. Biol. 5 (1): 27–40.

doi:10.1089/cmb.1998.5.27. PMID 9541869.

P versus NP problem 16

[15] Cook, Stephen (April 2000). The P versus NP Problem (http:/ / www. claymath. org/ millennium/ P_vs_NP/ Official_Problem_Description.pdf). Clay Mathematics Institute. . Retrieved 2006-10-18.

[16] L. R. Foulds (October 1983). "The Heuristic Problem-Solving Approach" (http:/ / www. jstor. org/ pss/ 2580891). The Journal of theOperational Research Society 34 (10): 927–934. doi:10.2307/2580891. .

[17] R. Impagliazzo, "A personal view of average-case complexity," (http:/ / cseweb. ucsd. edu/ ~russell/ average. ps) sct, pp.134, 10th AnnualStructure in Complexity Theory Conference (SCT'95), 1995

[18] http:/ / intractability. princeton. edu/ blog/ 2009/ 05/ program-for-workshop-on-impagliazzos-worlds/[19] T. P. Baker, J. Gill, R. Solovay. Relativizations of the P =? NP Question. SIAM Journal on Computing, 4(4): 431–442 (1975)[20] S. Aaronson and A. Wigderson. Algebrization: A New Barrier in Complexity Theory, in Proceedings of ACM STOC'2008, pp. 731–740.[21] Aaronson, Scott. "Is P Versus NP Formally Independent?" (http:/ / www. scottaaronson. com/ papers/ pnp. pdf). .[22] Ben-David, Shai; Halevi, Shai (1992). On the independence of P versus NP (http:/ / www. cs. technion. ac. il/ ~shai/ ph. ps. gz). Technical

Report. 714. Technion. .[23] John Markoff (8 October 2009). "Prizes Aside, the P-NP Puzzler Has Consequences" (http:/ / www. nytimes. com/ 2009/ 10/ 08/ science/

Wpolynom. html). The New York Times. .[24] Gerhard J. Woeginger (2010-08-09). "The P-versus-NP page" (http:/ / www. win. tue. nl/ ~gwoegi/ P-versus-NP. htm). . Retrieved

2010-08-12.[25] Markoff, John (16 August 2010). "Step 1: Post Elusive Proof. Step 2: Watch Fireworks." (http:/ / www. nytimes. com/ 2010/ 08/ 17/ science/

17proof. html?_r=1). The New York Times. . Retrieved 20 September 2010.[26] Polymath project wiki. "Deolalikar's P vs NP paper" (http:/ / michaelnielsen. org/ polymath1/ index. php?title=Deolalikar_P_vs_NP_paper).

.[27] Science News, "Crowdsourcing peer review" (http:/ / www. sciencenews. org/ index/ generic/ activity/ view/ id/ 63252/ title/

Crowdsourcing_peer_review)[28] Dick Lipton (12 August 2010). "Fatal Flaws in Deolalikar's Proof?" (http:/ / rjlipton. wordpress. com/ 2010/ 08/ 12/

fatal-flaws-in-deolalikars-proof/ ). .[29] Dick Lipton (15 September 2010). "An Update on Vinay Deolalikar's Proof" (http:/ / rjlipton. wordpress. com/ 2010/ 09/ 15/

an-update-on-vinay-deolalikars-proof/ ). . Retrieved December 31, 2010.[30] M. Agrawal, N. Kayal, N. Saxena. "Primes is in P" (http:/ / www. cse. iitk. ac. in/ users/ manindra/ algebra/ primality_v6. pdf) (PDF). .

Retrieved 2008-12-29.[31] AKS primality test

Further reading• Fraenkel, A. S.; Lichtenstein, D.. Computing a Perfect Strategy for n*n Chess Requires Time Exponential in N.

(http:/ / www. pubzone. org/ dblp/ conf/ icalp/ FraenkelL81). doi:10.1007/3-540-10843-2+23.• Garey, Michael (1979). Computers and Intractability. San Francisco: W.H. Freeman. ISBN 0716710455.• Goldreich, Oded (2010). P, Np, and Np-Completeness. Cambridge: Cambridge University Press.

ISBN 9780521122542.• Immerman, N. (1983). Languages which capture complexity classes. pp. 347. doi:10.1145/800061.808765.• Cormen, Thomas (2001). Introduction to Algorithms. Cambridge: MIT Press. ISBN 0262032937.• Papadimitriou, Christos (1994). Computational Complexity. Boston: Addison-Wesley. ISBN 0201530821.• Fortnow, L. (2009). "The status of the P versus NP problem". Communications of the ACM 52 (9): 78.

doi:10.1145/1562164.1562186.

External links• The Clay Mathematics Institute Millennium Prize Problems (http:/ / www. claymath. org/ millennium/ )• The Clay Math Institute Official Problem Description (http:/ / www. claymath. org/ millennium/ P_vs_NP/

Official_Problem_Description. pdf)PDF (118 KB)• Ian Stewart on Minesweeper as NP-complete at The Clay Math Institute (http:/ / www. claymath. org/

Popular_Lectures/ Minesweeper/ )• Gerhard J. Woeginger. The P-versus-NP page (http:/ / www. win. tue. nl/ ~gwoegi/ P-versus-NP. htm). A list of

links to a number of purported solutions to the problem. Some of these links state that P equals NP, some of themstate the opposite. It is probable that all these alleged solutions are incorrect.

• Computational Complexity of Games and Puzzles (http:/ / www. ics. uci. edu/ ~eppstein/ cgt/ hard. html)

P versus NP problem 17

• Complexity Zoo: Class P (http:/ / qwiki. stanford. edu/ index. php/ Complexity_Zoo:P#p), Complexity Zoo: ClassNP (http:/ / qwiki. stanford. edu/ index. php/ Complexity_Zoo:N#np)

• Scott Aaronson 's Shtetl Optimized blog: Reasons to believe (http:/ / scottaaronson. com/ blog/ ?p=122), a list ofjustifications for the belief that P ≠ NP

Hodge conjectureThe Hodge conjecture is a major unsolved problem in algebraic geometry which relates the algebraic topology of anon-singular complex algebraic variety and the subvarieties of that variety. More specifically, the conjecture saysthat certain de Rham cohomology classes are algebraic, that is, they are sums of Poincaré duals of the homologyclasses of subvarieties. The Hodge conjecture is one of the Clay Mathematics Institute's Millennium Prize Problems.

Millennium Prize Problems

P versus NP problem

Hodge conjecture

Poincaré conjecture (solution)

Riemann hypothesis

Yang–Mills existence and mass gap

Navier–Stokes existence and smoothness

Birch and Swinnerton-Dyer conjecture

MotivationLet X be a compact complex manifold of complex dimension n. Then X is an orientable smooth manifold of realdimension 2n, so its cohomology groups lie in degrees zero through 2n. X is a Kähler manifold, so that there is adecomposition on its cohomology with complex coefficients:

where is the subgroup of cohomology classes which are represented by harmonic forms of type (p, q).That is, these are the cohomology classes represented by differential forms which, in some choice of localcoordinates , can be written as a harmonic function times .(See Hodge theory for more details.) Taking wedge products of these harmonic representatives corresponds to thecup product in cohomology, so the cup product is compatible with the Hodge decomposition:

Since X is a compact oriented manifold, X has a fundamental class.Let Z be a complex submanifold of X of dimension k, and let i : Z → X be the inclusion map. Choose a differentialform of type (p, q). We can integrate over Z:

To evaluate this integral, choose a point of Z and call it 0. Around 0, we can choose local coordinates on X such that Z is just . If p > k, then must contain some where pulls back tozero on Z. The same is true if q > k. Consequently, this integral is zero if (p, q) ≠ (k, k).More abstractly, the integral can be written as the cap product of the homology class of Z and the cohomology class represented by . By Poincaré duality, the homology class of Z is dual to a cohomology class which we will call

Hodge conjecture 18

[Z], and the cap product can be computed by taking the cup product of [Z] and and capping with the fundamentalclass of X. Because [Z] is a cohomology class, it has a Hodge decomposition. By the computation we did above, ifwe cup this class with any class of type (p, q) ≠ (k, k), then we get zero. Because , we conclude that [Z] must lie in .Loosely speaking, the Hodge conjecture asks:

Which cohomology classes in come from complex subvarieties Z?

Statement of the Hodge conjectureLet:

We call this the group of Hodge classes of degree 2k on X.The modern statement of the Hodge conjecture is:

Hodge conjecture. Let X be a projective complex manifold. Then every Hodge class on X is a linearcombination with rational coefficients of the cohomology classes of complex subvarieties of X.

A projective complex manifold is a complex manifold which can be embedded in complex projective space. Becauseprojective space carries a Kähler metric, the Fubini-Study metric, such a manifold is always a Kähler manifold. ByChow's theorem, a projective complex manifold is also a smooth projective algebraic variety, that is, it is the zero setof a collection of homogenous polynomials.

Reformulation in terms of algebraic cyclesAnother way of phrasing the Hodge conjecture involves the idea of an algebraic cycle. An algebraic cycle on X is aformal combination of subvarieties of X, that is, it is something of the form:

The coefficients are usually taken to be integral or rational. We define the cohomology class of an algebraic cycle tobe the sum of the cohomology classes of its components. This is an example of the cycle class map of de Rhamcohomology, see Weil cohomology. For example, the cohomology class of the above cycle would be:

Such a cohomology class is called algebraic. With this notation, the Hodge conjecture becomes:Let X be a projective complex manifold. Then every Hodge class on X is algebraic.

Known cases of the Hodge conjecture

Low dimension and codimensionThe first result on the Hodge conjecture is due to Solomon Lefschetz. In fact, it predates the conjecture and providedsome of Hodge's motivation.

Theorem (Lefschetz theorem on (1,1)-classes) Any element of is the cohomologyclass of a divisor on X. In particular, the Hodge conjecture is true for .

A very quick proof can be given using sheaf cohomology and the exponential exact sequence. (The cohomologyclass of a divisor turns out to equal to its first Chern class.) Lefschetz's original proof proceeded by normal functions,which were introduced by Henri Poincaré. However, Griffiths's transversality theorem shows that this approachcannot prove the Hodge conjecture for higher codimensional subvarieties.By the Hard Lefschetz theorem, one can prove:

Hodge conjecture 19

Theorem. If the Hodge conjecture holds for Hodge classes of degree p, p < n, then the Hodge conjecture holdsfor Hodge classes of degree 2n − p.

Combining the above two theorems implies that Hodge conjecture is true for Hodge classes of degree 2n − 2. Thisproves the Hodge conjecture when X has dimension at most three.The Lefschetz theorem on (1,1)-classes also implies that if all Hodge classes are generated by the Hodge classes ofdivisors, then the Hodge conjecture is true:

Corollary. If the algebra

is generated by Hdg1(X), then the Hodge conjecture holds for X.

Abelian varieties

For most abelian varieties, the algebra is generated in degree one, so the Hodge conjecture holds. Inparticular, the Hodge conjecture holds for sufficiently general abelian varieties, for products of elliptic curves, andfor simple abelian varieties. However, David Mumford constructed an example of an abelian variety where

is not generated by products of divisor classes. André Weil generalized this example by showing thatwhenever the variety has complex multiplication by an imaginary quadratic field, then is not generatedby products of divisor classes. Moonen and Zarhin proved that in dimension less than 5, either isgenerated in degree one, or the variety has complex multiplication by an imaginary quadratic field. In the latter case,the Hodge conjecture is only known in special cases.

Generalizations

The integral Hodge conjectureHodge's original conjecture was:

Integral Hodge conjecture. Let X be a projective complex manifold. Then every cohomology class inis the cohomology class of an algebraic cycle with integral coefficients on X.

This is now known to be false. The first counterexample was constructed by Michael Atiyah and FriedrichHirzebruch. Using K-theory, they constructed an example of a torsion Hodge class, that is, a Hodge class suchthat for some positive integer n, . Such a cohomology class cannot be the class of a cycle. Burt Totaroreinterpreted their result in the framework of cobordism and found many examples of torsion classes.The simplest adjustment of the integral Hodge conjecture is:

Integral Hodge conjecture modulo torsion. Let X be a projective complex manifold. Then every non-torsioncohomology class in is the cohomology class of an algebraic cycle with integralcoefficients on X.

This is also false. János Kollár found an example of a Hodge class which is not algebraic, but which has anintegral multiple which is algebraic.

Hodge conjecture 20

The Hodge conjecture for Kähler varietiesA natural generalization of the Hodge conjecture would ask:

Hodge conjecture for Kähler varieties, naive version. Let X be a complex Kähler manifold. Then everyHodge class on X is a linear combination with rational coefficients of the cohomology classes of complexsubvarieties of X.

This is too optimistic, because there are not enough subvarieties to make this work. A possible substitute is to askinstead one of the two following questions:

Hodge conjecture for Kähler varieties, vector bundle version. Let X be a complex Kähler manifold. Thenevery Hodge class on X is a linear combination with rational coefficients of Chern classes of vector bundles onX.Hodge conjecture for Kähler varieties, coherent sheaf version. Let X be a complex Kähler manifold. Thenevery Hodge class on X is a linear combination with rational coefficients of Chern classes of coherent sheaveson X.

Claire Voisin proved that the Chern classes of coherent sheaves give strictly more Hodge classes than the Chernclasses of vector bundles and that the Chern classes of coherent sheaves are insufficient to generate all the Hodgeclasses. Consequently, the only known formulations of the Hodge conjecture for Kähler varieties are false.

The generalized Hodge conjectureHodge made an additional, stronger conjecture than the integral Hodge conjecture. Say that a cohomology class on Xis of level c if it is the pushforward of a cohomology class on a c-codimensional subvariety of X. The cohomologyclasses of level at least c filter the cohomology of X, and it is easy to see that the cth step of the filtration

satisfies

Hodge's original statement was:Generalized Hodge conjecture, Hodge's version.

Grothendieck observed that this cannot be true, even with rational coefficients, because the right-hand side is notalways a Hodge structure. His corrected form of the Hodge conjecture is:

Generalized Hodge conjecture. is the largest sub-Hodge structure of containedin

This version is open.

Algebraicity of Hodge lociThe strongest evidence in favor of the Hodge conjecture is the algebraicity result of Cattani, Deligne and Kaplan.Suppose that we vary the complex structure of X over a simply connected base. Then the topological cohomology ofX does not change, but the Hodge decomposition does change. It is known that if the Hodge conjecture is true, thenthe locus of all points on the base where the cohomology of a fiber is a Hodge class is in fact an algebraic subset, thatis, it is cut out by polynomial equations. Cattani, Deligne, and Kaplan proved that this is always true, withoutassuming the Hodge conjecture.

Hodge conjecture 21

References• Cattani, Eduardo; Deligne, Pierre; Kaplan, Aroldo (1995), "On the locus of Hodge classes" [1], Journal of the

American Mathematical Society (American Mathematical Society) 8 (2): 483–506, doi:10.2307/2152824,ISSN 0894-0347, MR1273413

• Hodge, W. V. D. (1950), "The topological invariants of algebraic varieties", Proceedings of the InternationalCongress of Mathematicians (Cambridge, MA) 1: 181–192.

• Grothendieck, A (1969), "Hodge's general conjecture is false for trivial reasons", Topology 8: 299–303,doi:10.1016/0040-9383(69)90016-0.

External links• The Clay Math Institute Official Problem Description (pdf) [2]

• Popular lecture on Hodge Conjecture by Dan Freed (University of Texas) (Real Video) [3] (Slides) [4]

• Indranil Biswas; Kapil Paranjape. The Hodge Conjecture for general Prym varieties [5]

References[1] http:/ / jstor. org/ stable/ 2152824[2] http:/ / www. claymath. org/ millennium/ Hodge_Conjecture/ Official_Problem_Description. pdf[3] http:/ / claymath. msri. org/ hodgeconjecture. mov[4] http:/ / www. ma. utexas. edu/ users/ dafr/ HodgeConjecture/ netscape_noframes. html[5] http:/ / arxiv. org/ abs/ math/ 0007192v1

Riemann hypothesis

The real part (red) and imaginary part (blue) ofthe Riemann zeta function along the critical line

Re(s) = 1/2. The first non-trivial zeros can beseen at Im(s) = ±14.135, ±21.022 and ±25.011.

Riemann hypothesis 22

Millennium Prize Problems

P versus NP problem

Hodge conjecture

Poincaré conjecture (solution)

Riemann hypothesis

Yang–Mills existence and mass gap

Navier–Stokes existence and smoothness

Birch and Swinnerton-Dyer conjecture

In mathematics, the Riemann hypothesis, proposed by Bernhard Riemann (1859), is a conjecture about thedistribution of the zeros of the Riemann zeta function which states that all non-trivial zeros have real part 1/2. Thename is also used for some closely related analogues, such as the Riemann hypothesis for curves over finite fields.The Riemann hypothesis implies results about the distribution of prime numbers that are in some ways as good aspossible. Along with suitable generalizations, it is considered by some mathematicians to be the most importantunresolved problem in pure mathematics (Bombieri 2000). The Riemann hypothesis is part of Problem 8, along withthe Goldbach conjecture, in Hilbert's list of 23 unsolved problems, and is also one of the Clay Mathematics InstituteMillennium Prize Problems. Since it was formulated, it has withstood concentrated efforts from many outstandingmathematicians. In 1973, Pierre Deligne proved an analogue of the Riemann Hypothesis for zeta functions ofvarieties defined over finite fields. The full version of the hypothesis remains unsolved, although modern computercalculations have shown that the first 10 trillion zeros lie on the critical line.The Riemann zeta function ζ(s) is defined for all complex numbers s ≠ 1. It has zeros at the negative even integers(i.e. at s = −2, −4, −6, ...). These are called the trivial zeros. The Riemann hypothesis is concerned with thenon-trivial zeros, and states that:

The real part of any non-trivial zero of the Riemann zeta function is 1/2.Thus the non-trivial zeros should lie on the critical line, 1/2 + it, where t is a real number and i is the imaginary unit.There are several popular books on the Riemann hypothesis, such as Derbyshire (2003), Rockmore (2005), Sabbagh(2003), du Sautoy (2003). The books Edwards (1974), Patterson (1988) and Borwein et al. (2008) give mathematicalintroductions, while Titchmarsh (1986), Ivić (1985) and Karatsuba & Voronin (1992) are advanced monographs.

The Riemann zeta functionThe Riemann zeta function is defined for complex s with real part greater than 1 by the absolutely convergentinfinite series

Leonhard Euler showed that this series equals the Euler product

where the infinite product extends over all prime numbers p, and again converges for complex s with real partgreater than 1. The convergence of the Euler product shows that ζ(s) has no zeros in this region, as none of thefactors have zeros.The Riemann hypothesis discusses zeros outside the region of convergence of this series, so it needs to beanalytically continued to all complex s. This can be done by expressing it in terms of the Dirichlet eta function asfollows. If s is greater than one, then the zeta function satisfies

Riemann hypothesis 23

However, the series on the right converges not just when s is greater than one, but more generally whenever s haspositive real part. Thus, this alternative series extends the zeta function from Re(s) > 1 to the larger domain Re(s) >0.In the strip 0 < Re(s) < 1 the zeta function also satisfies the functional equation

One may then define ζ(s) for all remaining nonzero complex numbers s by assuming that this equation holds outsidethe strip as well, and letting ζ(s) equal the right-hand side of the equation whenever s has non-positive real part. If sis a negative even integer then ζ(s) = 0 because the factor sin(πs/2) vanishes; these are the trivial zeros of the zetafunction. (If s is a positive even integer this argument does not apply because the zeros of sin are cancelled by thepoles of the gamma function.) The value ζ(0) = −1/2 is not determined by the functional equation, but is the limitingvalue of ζ(s) as s approaches zero. The functional equation also implies that the zeta function has no zeros withnegative real part other than the trivial zeros, so all non-trivial zeros lie in the critical strip where s has real partbetween 0 and 1.

History"…es ist sehr wahrscheinlich, dass alle Wurzeln reell sind. Hiervon wäre allerdings ein strenger Beweis zu wünschen; ich habeindess die Aufsuchung desselben nach einigen flüchtigen vergeblichen Versuchen vorläufig bei Seite gelassen, da er für dennächsten Zweck meiner Untersuchung entbehrlich schien.""…it is very probable that all roots are real. Of course one would wish for a rigorous proof here; I have for the time being, after somefleeting vain attempts, provisionally put aside the search for this, as it appears dispensable for the next objective of myinvestigation."

Riemann's statement of the Riemann hypothesis, from (Riemann 1859). (He was discussing a version of the zeta function, modifiedso that its roots are real rather than on the critical line.)

In his 1859 paper On the Number of Primes Less Than a Given Magnitude Riemann found an explicit formula for thenumber of primes π(x) less than a given number x. His formula was given in terms of the related function

which counts primes where a prime power pn counts as 1/n of a prime. The number of primes can be recovered fromthis function by

where μ is the Möbius function. Riemann's formula is then

where the sum is over the nontrivial zeros of the zeta function and where Π0 is a slightly modified version of Π thatreplaces its value at its points of discontinuity by the average of its upper and lower limits:

The summation in Riemann's formula is not absolutely convergent, but may be evaluated by taking the zeros ρ inorder of the absolute value of their imaginary part. The function Li occurring in the first term is the (unoffset)logarithmic integral function given by the Cauchy principal value of the divergent integral

Riemann hypothesis 24

The terms Li(xρ) involving the zeros of the zeta function need some care in their definition as Li has branch points at0 and 1, and are defined (for x > 1) by analytic continuation in the complex variable ρ in the region Re(ρ) > 0, i.e.they should be considered as Ei(ρ ln x). The other terms also correspond to zeros: the dominant term Li(x) comesfrom the pole at s = 1, considered as a zero of multiplicity −1, and the remaining small terms come from the trivialzeros. For some graphs of the sums of the first few terms of this series see Riesel & Göhl (1970) or Zagier (1977).This formula says that the zeros of the Riemann zeta function control the oscillations of primes around their"expected" positions. Riemann knew that the non-trivial zeros of the zeta function were symmetrically distributedabout the line s = 1/2 + it, and he knew that all of its non-trivial zeros must lie in the range 0 ≤ Re(s) ≤ 1. He checkedthat a few of the zeros lay on the critical line with real part 1/2 and suggested that they all do; this is the Riemannhypothesis.

Consequences of the Riemann hypothesisThe practical uses of the Riemann hypothesis include many propositions which are known to be true under theRiemann hypothesis, and some which can be shown to be equivalent to the Riemann hypothesis.

Distribution of prime numbersRiemann's explicit formula for the number of primes less than a given number in terms of a sum over the zeros of theRiemann zeta function says that the magnitude of the oscillations of primes around their expected position iscontrolled by the real parts of the zeros of the zeta function. In particular the error term in the prime number theoremis closely related to the position of the zeros: for example, the supremum of real parts of the zeros is the infimum ofnumbers β such that the error is O(xβ) (Ingham 1932).Von Koch (1901) proved that the Riemann hypothesis is equivalent to the "best possible" bound for the error of theprime number theorem.A precise version of Koch's result, due to Schoenfeld (1976), says that the Riemann hypothesis is equivalent to

Growth of arithmetic functionsThe Riemann hypothesis implies strong bounds on the growth of many other arithmetic functions, in addition to theprimes counting function above.One example involves the Möbius function μ. The statement that the equation

is valid for every s with real part greater than 1/2, with the sum on the right hand side converging, is equivalent to theRiemann hypothesis. From this we can also conclude that if the Mertens function is defined by

then the claim that

for every positive ε is equivalent to the Riemann hypothesis (Titchmarsh 1986). (For the meaning of these symbols, see Big O notation.) The determinant of the order n Redheffer matrix is equal to M(n), so the Riemann hypothesis can also be stated as a condition on the growth of these determinants. The Riemann hypothesis puts a rather tight

Riemann hypothesis 25

bound on the growth of M, since Odlyzko & te Riele (1985) disproved the slightly stronger Mertens conjecture

The Riemann hypothesis is equivalent to many other conjectures about the rate of growth of other arithmeticfunctions aside from μ(n). A typical example is Robin's theorem (Robin 1984), which states that if σ(n) is the divisorfunction, given by

then

for all n > 5040 if and only if the Riemann hypothesis is true, where γ is the Euler–Mascheroni constant.Another example was found by Franel & Landau (1924) showing that the Riemann hypothesis is equivalent to astatement that the terms of the Farey sequence are fairly regular. More precisely, if Fn is the Farey sequence of ordern, beginning with 1/n and up to 1/1, then the claim that for all ε > 0

is equivalent to the Riemann hypothesis. Here is the number of terms in the Farey sequence of order

n.For an example from group theory, if g(n) is Landau's function given by the maximal order of elements of thesymmetric group Sn of degree n, then Massias, Nicolas & Robin (1988) showed that the Riemann hypothesis isequivalent to the bound

for all sufficiently large n.

Lindelöf hypothesis and growth of the zeta functionThe Riemann hypothesis has various weaker consequences as well; one is the Lindelöf hypothesis on the rate ofgrowth of the zeta function on the critical line, which says that, for any ε > 0,

as t tends to infinity.The Riemann hypothesis also implies quite sharp bounds for the growth rate of the zeta function in other regions ofthe critical strip. For example, it implies that

so the growth rate of ζ(1+it) and its inverse would be known up to a factor of 2 (Titchmarsh 1986).

Large prime gap conjectureThe prime number theorem implies that on average, the gap between the prime p and its successor is log p. However, some gaps between primes may be much larger than the average. Cramér proved that, assuming the Riemann hypothesis, every gap is O(√p log p). This is a case when even the best bound that can currently be proved using the Riemann hypothesis is far weaker than what seems to be true: Cramér's conjecture implies that every gap is O((log p)2) which, while larger than the average gap, is far smaller than the bound implied by the Riemann

Riemann hypothesis 26

hypothesis. Numerical evidence supports Cramér's conjecture (Nicely 1999).

Criteria equivalent to the Riemann hypothesisMany statements equivalent to the Riemann hypothesis have been found, though so far none of them have led tomuch progress in solving it. Some typical examples are as follows.The Riesz criterion was given by Riesz (1916), to the effect that the bound

holds for all if and only if the Riemann hypothesis holds.Nyman (1950) proved that the Riemann Hypothesis is true if and only if the space of functions of the form

where ρ(z) is the fractional part of z, 0 ≤ θν ≤ 1, and

,

is dense in the Hilbert space L2(0,1) of square-integrable functions on the unit interval. Beurling (1955) extended thisby showing that the zeta function has no zeros with real part greater than 1/p if and only if this function space isdense in Lp(0,1)Salem (1953) showed that the Riemann hypothesis is true if and only if the integral equation

has no non-trivial bounded solutions φ for 1/2<σ<1.Weil's criterion is the statement that the positivity of a certain function is equivalent to the Riemann hypothesis.Related is Li's criterion, a statement that the positivity of a certain sequence of numbers is equivalent to the Riemannhypothesis.

Speiser (1934) proved that the Riemann hypothesis is equivalent to the statement that , the derivative of , has no zeros in the strip

That ζ has only simple zeros on the critical line is equivalent to its derivative having no zeros on the critical line.

Consequences of the generalized Riemann hypothesisSeveral applications use the generalized Riemann hypothesis for Dirichlet L-series or zeta functions of number fieldsrather than just the Riemann hypothesis. Many basic properties of the Riemann zeta function can easily begeneralized to all Dirichlet L-series, so it is plausible that a method that proves the Riemann hypothesis for theRiemann zeta function would also work for the generalized Riemann hypothesis for Dirichlet L-functions. Severalresults first proved using the generalized Riemann hypothesis were later given unconditional proofs without using it,though these were usually much harder. Many of the consequences on the following list are taken from Conrad(2010).• In 1913, Gronwall showed that the generalized Riemann hypothesis implies that Gauss's list of imaginary

quadratic fields with class number 1 is complete, though Baker, Stark and Heegner later gave unconditionalproofs of this without using the generalized Riemann hypothesis.

• In 1917, Hardy and Littlewood showed that the generalized Riemann hypothesis implies a conjecture ofChebyshev that

Riemann hypothesis 27

which says that in some sense primes 3 mod 4 are more common than primes 1 mod 4.• In 1923 Hardy and Littlewood showed that the generalized Riemann hypothesis implies a weak form of the

Goldbach conjecture for odd numbers: that every sufficiently large odd number is the sum of 3 primes, though in1937 Vinogradov gave an unconditional proof. In 1997 Deshouillers, Effinger, te Riele, and Zinoviev showed thatthe generalized Riemann hypothesis implies that every odd number greater than 5 is the sum of 3 primes.

• In 1934, Chowla showed that the generalized Riemann hypothesis implies that the first prime in the arithmeticprogression a mod m is at most Km2log(m)2 for some fixed constant K.

• In 1967, Hooley showed that the generalized Riemann hypothesis implies Artin's conjecture on primitive roots.• In 1973, Weinberger showed that the generalized Riemann hypothesis implies that Euler's list of idoneal numbers

is complete.• Weinberger (1973) showed that the generalized Riemann hypothesis for the zeta functions of all algebraic number

fields implies that any number field with class number 1 is either Euclidean or an imaginary quadratic numberfield of discriminant −19, −43, −67, or −163.

• In 1976, G. Miller showed that the generalized Riemann hypothesis implies that one can test if a number is primein polynomial times. In 2002, Manindra Agrawal, Neeraj Kayal and Nitin Saxena proved this resultunconditionally using the AKS primality test.

• Odlyzko (1990) discussed how the generalized Riemann hypothesis can be used to give sharper estimates fordiscriminants and class numbers of number fields.

• Ono & Soundararajan (1997) showed that the generalized Riemann hypothesis implies that Ramanujan's integralquadratic form x2 +y2 + 10z2 represents all integers that it represents locally, with exactly 18 exceptions.

Generalizations and analogues of the Riemann hypothesis

Dirichlet L-series and other number fieldsThe Riemann hypothesis can be generalized by replacing the Riemann zeta function by the formally similar, butmuch more general, global L-functions. In this broader setting, one expects the non-trivial zeros of the globalL-functions to have real part 1/2. It is these conjectures, rather than the classical Riemann hypothesis only for thesingle Riemann zeta function, which accounts for the true importance of the Riemann hypothesis in mathematics.The generalized Riemann hypothesis extends the Riemann hypothesis to all Dirichlet L-functions. In particular itimplies the conjecture that Siegel zeros (zeros of L functions between 1/2 and 1) do not exist.The extended Riemann hypothesis extends the Riemann hypothesis to all Dedekind zeta functions of algebraicnumber fields. The extended Riemann hypothesis for abelian extension of the rationals is equivalent to thegeneralized Riemann hypothesis. The Riemann hypothesis can also be extended to the L-functions of Heckecharacters of number fields.The grand Riemann hypothesis extends it to all automorphic zeta functions, such as Mellin transforms of Heckeeigenforms.

Riemann hypothesis 28

Function fields and zeta functions of varieties over finite fieldsArtin (1924) introduced global zeta functions of (quadratic) function fields and conjectured an analogue of theRiemann hypothesis for them, which has been proven by Hasse in the genus 1 case and by Weil (1948) in general.For instance, the fact that the Gauss sum, of the quadratic character of a finite field of size q (with q odd), hasabsolute value

is actually an instance of the Riemann hypothesis in the function field setting. This led Weil (1949) to conjecture asimilar statement for all algebraic varieties; the resulting Weil conjectures were proven by Pierre Deligne (1974,1980).

Selberg zeta functionsSelberg (1956) introduced the Selberg zeta function of a Riemann surface. These are similar to the Riemann zetafunction: they have a functional equation, and an infinite product similar to the Euler product but taken over closedgeodesics rather than primes. The Selberg trace formula is the analogue for these functions of the explicit formulas inprime number theory. Selberg proved that the Selberg zeta functions satisfy the analogue of the Riemann hypothesis,with the imaginary parts of their zeros related to the eigenvalues of the Laplacian operator of the Riemann surface.

Ihara zeta functionsThe Ihara zeta function of a finite graph is an analogue of the Selberg zeta function introduced by Yasutaka Ihara. Aregular finite graph is a Ramanujan graph, a mathematical model of efficient communication networks, if and only ifits Ihara zeta function satisfies the analogue of the Riemann hypothesis as was pointed out by T. Sunada.

Montgomery's pair correlation conjectureMontgomery (1973) suggested the pair correlation conjecture that the correlation functions of the (suitablynormalized) zeros of the zeta function should be the same as those of the eigenvalues of a random hermitian matrix.Odlyzko (1987) showed that this is supported by large scale numerical calculations of these correlation functions.Montgomery showed that (assuming the Riemann hypothesis) at least 2/3 of all zeros are simple, and a relatedconjecture is that all zeros of the zeta function are simple (or more generally have no non-trivial integer linearrelations between their imaginary parts). Dedekind zeta functions of algebraic number fields, which generalize theRiemann zeta function, often do have multiple complex zeros. This is because the Dedekind zeta functions factorizeas a product of powers of Artin L-functions, so zeros of Artin L-functions sometimes give rise to multiple zeros ofDedekind zeta functions. Other examples of zeta functions with multiple zeros are the L-functions of some ellipticcurves: these can have multiple zeros at the real point of their critical line; the Birch-Swinnerton-Dyer conjecturepredicts that the multiplicity of this zero is the rank of the elliptic curve.

Other zeta functionsThere are many other examples of zeta functions with analogues of the Riemann hypothesis, some of which havebeen proved. Goss zeta functions of function fields have a Riemann hypothesis, proved by Sheats (1998). The mainconjecture of Iwasawa theory, proved by Barry Mazur and Andrew Wiles for cyclotomic fields, and Wiles for totallyreal fields, identifies the zeros of a p-adic L-function with the eigenvalues of an operator, so can be thought of as ananalogue of the Hilbert–Pólya conjecture for p-adic L-functions (Wiles 2000).

Riemann hypothesis 29

Attempts to prove the Riemann hypothesisSeveral mathematicians have addressed the Riemann hypothesis, but none of their attempts have yet been acceptedas correct solutions. Watkins (2007) lists some incorrect solutions, and more are frequently announced [1].

Operator theoryHilbert and Polya suggested that one way to derive the Riemann hypothesis would be to find a self-adjoint operator,from the existence of which the statement on the real parts of the zeros of ζ(s) would follow when one applies thecriterion on real eigenvalues. Some support for this idea comes from several analogues of the Riemann zetafunctions whose zeros correspond to eigenvalues of some operator: the zeros of a zeta function of a variety over afinite field correspond to eigenvalues of a Frobenius element on an etale cohomology group, the zeros of a Selbergzeta function are eigenvalues of a Laplacian operator of a Riemann surface, and the zeros of a p-adic zeta functioncorrespond to eigenvectors of a Galois action on ideal class groups.Odlyzko (1987) showed that the distribution of the zeros of the Riemann zeta function shares some statisticalproperties with the eigenvalues of random matrices drawn from the Gaussian unitary ensemble. This gives somesupport to the Hilbert–Pólya conjecture.

In 1999, Michael Berry and Jon Keating conjectured that there is some unknown quantization of the classicalHamiltonian so that

and even more strongly, that the Riemann zeros coincide with the spectrum of the operator . This is tobe contrasted to canonical quantization which leads to the Heisenberg uncertainty principle and thenatural numbers as spectrum of the quantum harmonic oscillator. The crucial point is that the Hamiltonian should bea self-adjoint operator so that the quantization would be a realization of the Hilbert–Pólya program. In a connectionwith this quantum mechanical problem Berry and Connes had proposed that the inverse of the potential of theHamiltonian is connected to the half-derivative of the function then, in

Berry-Connes approach (Connes 1999). This yields to a Hamiltonian whose

eigenvalues are the square of the imaginary part of the Riemann zeros, also the functional determinant of thisHamiltonian operator is just the Riemann Xi-functionThe analogy with the Riemann hypothesis over finite fields suggests that the Hilbert space containing eigenvectorscorresponding to the zeros might be some sort of first cohomology group of the spectrum Spec(Z) of the integers.Deninger (1998) described some of the attempts to find such a cohomology theory.Zagier (1983) constructed a natural space of invariant functions on the upper half plane which has eigenvalues underthe Laplacian operator corresponding to zeros of the Riemann zeta function, and remarked that in the unlikely eventthat one could show the existence of a suitable positive definite inner product on this space the Riemann hypothesiswould follow. Cartier (1982) discussed a related example, where due to a bizarre bug a computer program listedzeros of the Riemann zeta function as eigenvalues of the same Laplacian operator.Schumayer & Hutchinson (2011) surveyed some of the attempts to construct a suitable physical model related to theRiemann zeta function.

Riemann hypothesis 30

Lee–Yang theoremThe Lee–Yang theorem states that the zeros of certain partition functions in statistical mechanics all lie on a "criticalline" with real part 0, and this has led to some speculation about a relationship with the Riemann hypothesis (Knauf1999).

Turán's resultPál Turán (1948) showed that if the functions

have no zeros when the real part of s is greater than one then

for all x > 0,

where λ(n) is the Liouville function given by (−1)r if n has r prime factors. He showed that this in turn would implythat the Riemann hypothesis is true. However Haselgrove (1958) proved that T(x) is negative for infinitely many x(and also disproved the closely related Polya conjecture), and Borwein, Ferguson & Mossinghoff (2008) showed thatthe smallest such x is 72185376951205. Spira (1968) showed by numerical calculation that the finite Dirichlet seriesabove for N=19 has a zero with real part greater than 1. Turán also showed that a somewhat weaker assumption, thenonexistence of zeros with real part greater than 1+N−1/2+ε for large N in the finite Dirichlet series above, would alsoimply the Riemann hypothesis, but Montgomery (1983) showed that for all sufficiently large N these series havezeros with real part greater than 1 + (log log N)/(4 log N). Therefore, Turán's result is vacuously true and cannot beused to help prove the Riemann hypothesis.

Noncommutative geometryConnes (1999, 2000) has described a relationship between the Riemann hypothesis and noncommutative geometry,and shows that a suitable analogue of the Selberg trace formula for the action of the idèle class group on the adèleclass space would imply the Riemann hypothesis. Some of these ideas are elaborated in Lapidus (2008).

Hilbert spaces of entire functionsLouis de Branges (1992) showed that the Riemann hypothesis would follow from a positivity condition on a certainHilbert space of entire functions. However Conrey & Li (2000) showed that the necessary positivity conditions arenot satisfied.

QuasicrystalsThe Riemann hypothesis implies that the zeros of the zeta function form a quasicrystal, meaning a distribution withdiscrete support whose Fourier transform also has discrete support. Dyson (2009) suggested trying to prove theRiemann hypothesis by classifying, or at least studying, 1-dimensional quasicrystals.

Multiple zeta functionsDeligne's proof of the Riemann hypothesis over finite fields used the zeta functions of product varieties, whose zerosand poles correspond to sums of zeros and poles of the original zeta function, in order to bound the real parts of thezeros of the original zeta function. By analogy, Kurokawa (1992) introduced multiple zeta functions whose zeros andpoles correspond to sums of zeros and poles of the Riemann zeta function. To make the series converge he restrictedto sums of zeros or poles all with non-negative imaginary part. So far, the known bounds on the zeros and poles ofthe multiple zeta functions are not strong enough to give useful estimates for the zeros of the Riemann zeta function.

Riemann hypothesis 31

Location of the zeros

Number of zerosThe functional equation combined with the argument principle implies that the number of zeros of the zeta functionwith imaginary part between 0 and T is given by

for s=1/2+iT, where the argument is defined by varying it continuously along the line with Im(s)=T, starting withargument 0 at ∞+iT. This is the sum of a large but well understood term

and a small but rather mysterious term

So the density of zeros with imaginary part near T is about log(T)/2π, and the function S describes the smalldeviations from this. The function S(t) jumps by 1 at each zero of the zeta function, and for t ≥ 8 it decreasesmonotonically between zeros with derivative close to −log t.

Karatsuba (1996) proved that every interval for contains at least

points where the function changes sign.Selberg (1946) showed that the average moments of even powers of S are given by

This suggests that S(T)/(log log T)1/2 resembles a Gaussian random variable with mean 0 and variance 2π2 (Ghosh(1983) proved this fact). In particular |S(T)| is usually somewhere around (log log T)1/2, but occasionally muchlarger. The exact order of growth of S(T) is not known. There has been no unconditional improvement to Riemann'soriginal bound S(T)=O(log T), though the Riemann hypothesis implies the slightly smaller bound S(T)=O(log T/loglog T) (Titchmarsh 1985). The true order of magnitude may be somewhat less than this, as random functions with thesame distribution as S(T) tend to have growth of order about log(T)1/2. In the other direction it cannot be too small:Selberg (1946) showed that S(T) ≠ o((log T)1/3/(log log T)7/3), and assuming the Riemann hypothesis Montgomeryshowed that S(T) ≠ o((log T)1/2/(log log T)1/2).Numerical calculations confirm that S grows very slowly: |S(T)| < 1 for T < 280, |S(T)| < 2 for T < 6800000, and thelargest value of |S(T)| found so far is not much larger than 3 (Odlyzko 2002).Riemann's estimate S(T) = O(log T) implies that the gaps between zeros are bounded, and Littlewood improved thisslightly, showing that the gaps between their imaginary parts tends to 0.

Riemann hypothesis 32

The theorem of Hadamard and de la Vallée-PoussinHadamard (1896) and de la Vallée-Poussin (1896) independently proved that no zeros could lie on the line Re(s) = 1.Together with the functional equation and the fact that there are no zeros with real part greater than 1, this showedthat all non-trivial zeros must lie in the interior of the critical strip 0 < Re(s) < 1. This was a key step in their firstproofs of the prime number theorem.Both the original proofs that the zeta function has no zeros with real part 1 are similar, and depend on showing that ifζ(1+it) vanishes, then ζ(1+2it) is singular, which is not possible. One way of doing this is by using the inequality

for σ>1, t real,and looking at the limit as σ tends to 1. This inequality follows by taking the real part of the log of the Euler productto see that

(where the sum is over all prime powers pn) so that

which is at least 1 because all the terms in the sum are positive, due to the inequality

Zero-free regionsDe la Vallée-Poussin (1899-1900) proved that if σ+it is a zero of the Riemann zeta function, then 1-σ ≥ C/log(t) forsome positive constant C. In other words zeros cannot be too close to the line σ=1: there is a zero-free region close tothis line. This zero-free region has been enlarged by several authors. Ford (2002) gave a version with explicitnumerical constants: ζ(σ + it) ≠ 0 whenever |t| ≥ 3 and

Zeros on the critical lineHardy (1914) and Hardy & Littlewood (1921) showed there are infinitely many zeros on the critical line, byconsidering moments of certain functions related to the zeta function. Selberg (1942) proved that at least a (small)positive proportion of zeros lie on the line. Levinson (1974) improved this to one-third of the zeros by relating thezeros of the zeta function to those of its derivative, and Conrey (1989) improved this further to two-fifths.Most zeros lie close to the critical line. More precisely, Bohr & Landau (1914) showed that for any positive ε, all butan infinitely small proportion of zeros lie within a distance ε of the critical line. Ivić (1985) gives several moreprecise versions of this result, called zero density estimates, which bound the number of zeros in regions withimaginary part at most T and real part at least 1/2+ε.

Riemann hypothesis 33

The Hardy-Littlewood conjectures

In 1914 Godfrey Harold Hardy proved that has infinitely many real zeros.Let be the total number of real zeros, be the total number of zeros of odd order of the function

, lying on the interval .The next two conjectures of Hardy and John Edensor Littlewood on the distance between real zeros of and on the density of zeros of on intervals for sufficiently great , and with as less as possible value of , where is an arbitrarily small number, open two new directionsin the investigation of the Riemann zeta function:1. for any there exists such that for and the interval

contains a zero of odd order of the function .2. for any there exist and , such that for and theinequality is true.

The Selberg conjectureAtle Selberg (1942) investigated the problem of Hardy-Littlewood 2 and proved that for any there exists such

and , such that for and the inequalityis true. Selberg conjectured that this could be tightened to . A.

A. Karatsuba (1984a, 1984b, 1985) proved that for a fixed satisfying the condition , asufficiently large and , , the interval contains at least

real zeros of the Riemann zeta function and therefore confirmed the Selberg conjecture. The estimates

of Selberg and Karatsuba can not be improved in respect of the order of growth as .Karatsuba (1992) proved that an analog of the Selberg conjecture holds for almost all intervals ,

, where is an arbitrarily small fixed positive number. The Karatsuba method permits to investigatezeros of the Riemann zeta-function on "supershort" intervals of the critical line, that is, on the intervals

, the length of which grows slower than any, even arbitrarily small degree . In particular, heproved that for any given numbers , satisfying the conditions almost all intervals

for contain at least zeros of the function . Thisestimate is quite close to the one that follows from the Riemann hypothesis.

Riemann hypothesis 34

Numerical calculations

Absolute value of the ζ-function

The function

has the same zeros as the zeta function in the critical strip, and is real on the critical line because of the functionalequation, so one can prove the existence of zeros exactly on the real line between two points by checkingnumerically that the function has opposite signs at these points. Usually one writes

where Hardy's function Z and the Riemann-Siegel theta function θ are uniquely defined by this and the condition thatthey are smooth real functions with θ(0)=0. By finding many intervals where the function Z changes sign one canshow that there are many zeros on the critical line. To verify the Riemann hypothesis up to a given imaginary part Tof the zeros, one also has to check that there are no further zeros off the line in this region. This can be done bycalculating the total number of zeros in the region and checking that it is the same as the number of zeros found onthe line. This allows one to verify the Riemann hypothesis computationally up to any desired value of T (provided allthe zeros of the zeta function in this region are simple and on the critical line).Some calculations of zeros of the zeta function are listed below. So far all zeros that have been checked are on thecritical line and are simple. (A multiple zero would cause problems for the zero finding algorithms, which depend onfinding sign changes between zeros.) For tables of the zeros, see Haselgrove & Miller (1960) or Odlyzko.

Year Number of zeros Author

1859? 3 B. Riemann used the Riemann-Siegel formula (unpublished, but reported in Siegel 1932).

1903 15 J. P. Gram (1903) used Euler–Maclaurin summation and discovered Gram's law. He showed that all 10 zeroswith imaginary part at most 50 range lie on the critical line with real part 1/2 by computing the sum of theinverse 10th powers of the roots he found.

1914 79 (γn ≤ 200) R. J. Backlund (1914) introduced a better method of checking all the zeros up to that point are on the line, bystudying the argument S(T) of the zeta function.

1925 138 (γn ≤ 300) J. I. Hutchinson (1925) found the first failure of Gram's law, at the Gram point g126.

1935 195 E. C. Titchmarsh (1935) used the recently rediscovered Riemann-Siegel formula, which is much faster thanEuler–Maclaurin summation.It takes about O(T3/2+ε) steps to check zeros with imaginary part less than T, whilethe Euler–Maclaurin method takes about O(T2+ε) steps.

Riemann hypothesis 35

1936 1041 E. C. Titchmarsh (1936) and L. J. Comrie were the last to find zeros by hand.

1953 1104 A. M. Turing (1953) found a more efficient way to check that all zeros up to some point are accounted for by thezeros on the line, by checking that Z has the correct sign at several consecutive Gram points and using the factthat S(T) has average value 0. This requires almost no extra work because the sign of Z at Gram points is alreadyknown from finding the zeros, and is still the usual method used. This was the first use of a digital computer tocalculate the zeros.

1956 15000 D. H. Lehmer (1956) discovered a few cases where the zeta function has zeros that are "only just" on the line:two zeros of the zeta function are so close together that it is unusually difficult to find a sign change betweenthem. This is called "Lehmer's phenomenon", and first occurs at the zeros with imaginary parts 7005.063 and7005.101, which differ by only .04 while the average gap between other zeros near this point is about 1.

1956 25000 D. H. Lehmer

1958 35337 N. A. Meller

1966 250000 R. S. Lehman

1968 3500000 Rosser, Yohe & Schoenfeld (1969) stated Rosser's rule (described below).

1977 40000000 R. P. Brent

1979 81000001 R. P. Brent

1982 200000001 R. P. Brent, J. van de Lune, H. J. J. te Riele, D. T. Winter

1983 300000001 J. van de Lune, H. J. J. te Riele

1986 1500000001 van de Lune, te Riele & Winter (1986) gave some statistical data about the zeros and give several graphs of Z atplaces where it has unusual behavior.

1987 A few of large (~1012)height

A. M. Odlyzko (1987) computed smaller numbers of zeros of much larger height, around 1012, to high precisionto check Montgomery's pair correlation conjecture.

1992 A few of large (~1020)height

A. M. Odlyzko (1992) computed a 175 million zeroes of heights around 1020 and a few more of heights around2×1020, and gave an extensive discussion of the results.

1998 10000 of large (~1021)height

A. M. Odlyzko (1998) computed some zeros of height about 1021

2001 10000000000 J. van de Lune (unpublished)

2004 900000000000 S. Wedeniwski (ZetaGrid distributed computing)

2004 10000000000000 and afew of large (up to~1024) heights

X. Gourdon (2004) and Patrick Demichel used the Odlyzko–Schönhage algorithm. They also checked twobillion zeros around heights 1013, 1014, ... , 1024.

Gram pointsA Gram point is a value of t such that ζ(1/2 + it) = Z(t)e − iθ(t) is a non-zero real; these are easy to find because theyare the points where the Euler factor at infinity π−s/2Γ(s/2) is real at s = 1/2 + it, or equivalently θ(t) is a multiple nπof π. They are usually numbered as gn for n = −1, 0, 1, ..., where gn is the unique solution of θ(t) = nπ with t ≥ 8 (θ isincreasing beyond this point; there is a second point with θ(t) = −π near 3.4, and θ(0) = 0). Gram observed that therewas often exactly one zero of the zeta function between any two Gram points; Hutchinson called this observationGram's law. There are several other closely related statements that are also sometimes called Gram's law: forexample, (−1)nZ(gn) is usually positive, or Z(t) usually has opposite sign at consecutive Gram points. The imaginaryparts γn of the first few zeros (in blue) and the first few Gram points gn are given in the following table

g−1 γ1 g0 γ2 g1 γ3 g2 γ4 g3 γ5 g4 γ6 g5

0 3.4 9.667 14.135 17.846 21.022 23.170 25.011 27.670 30.425 31.718 32.935 35.467 37.586 38.999

Riemann hypothesis 36

This shows the values of ζ(1/2+it) in the complexplane for 0 ≤ t ≤ 34. (For t=0, ζ(1/2) ≈ -1.460

corresponds to the leftmost point of the redcurve.) Gram's law states that the curve usually

crosses the real axis once between zeros.

The first failure of Gram's law occurs at the 127'th zero and the Grampoint g126, which are in the "wrong" order.

g124 γ126 g125 g126 γ127

γ128 g127 γ129 g128

279.148 279.229 280.802 282.455 282.465 283.211 284.104 284.836 285.752

A Gram point t is called good if the zeta function is positive at 1/2 + it. The indices of the "bad" Gram points whereZ has the "wrong" sign are 126, 134, 195, 211,... (sequence A114856 [2] in OEIS). A Gram block is an intervalbounded by two good Gram points such that all the Gram points between them are bad. A refinement of Gram's lawcalled Rosser's rule due to Rosser, Yohe & Schoenfeld (1969) says that Gram blocks often have the expected numberof zeros in them (the same as the number of Gram intervals), even though some of the individual Gram intervals inthe block may not have exactly one zero in them. For example, the interval bounded by g125 and g127 is a Gramblock containing a unique bad Gram point g126, and contains the expected number 2 of zeros although neither of itstwo Gram intervals contains a unique zero. Rosser et al. checked that there were no exceptions to Rosser's rule in thefirst 3 million zeros, although there are infinitely many exceptions to Rosser's rule over the entire zeta function.Gram's rule and Rosser's rule both say that in some sense zeros do not stray too far from their expected positions.The distance of a zero from its expected position is controlled by the function S defined above, which growsextremely slowly: its average value is of the order of (log log T)1/2, which only reaches 2 for T around 1024. Thismeans that both rules hold most of the time for small T but eventually break down often.

Arguments for and against the Riemann hypothesisMathematical papers about the Riemann hypothesis tend to be cautiously noncommittal about its truth. Of authorswho express an opinion, most of them, such as Riemann (1859) or Bombieri (2000), imply that they expect (or atleast hope) that it is true. The few authors who express serious doubt about it include Ivić (2008) who lists somereasons for being skeptical, and Littlewood (1962) who flatly states that he believes it to be false, and that there is noevidence whatever for it and no imaginable reason for it to be true. The consensus of the survey articles (Bombieri2000, Conrey 2003, and Sarnak 2008) is that the evidence for it is strong but not overwhelming, so that while it isprobably true there is some reasonable doubt about it.Some of the arguments for (or against) the Riemann hypothesis are listed by Sarnak (2008), Conrey (2003), and Ivić(2008), and include the following reasons.• Several analogues of the Riemann hypothesis have already been proved. The proof of the Riemann hypothesis for

varieties over finite fields by Deligne (1974) is possibly the single strongest theoretical reason in favor of the Riemann hypothesis. This provides some evidence for the more general conjecture that all zeta functions

Riemann hypothesis 37

associated with automorphic forms satisfy a Riemann hypothesis, which includes the classical Riemannhypothesis as a special case. Similarly Selberg zeta functions satisfy the analogue of the Riemann hypothesis, andare in some ways similar to the Riemann zeta function, having a functional equation and an infinite productexpansion analogous to the Euler product expansion. However there are also some major differences; for examplethey are not given by Dirichlet series. The Riemann hypothesis for the Goss zeta function was proved by Sheats(1998). In contrast to these positive examples, however, some Epstein zeta functions do not satisfy the Riemannhypothesis, even though they have an infinite number of zeros on the critical line (Titchmarsh 1986). Thesefunctions are quite similar to the Riemann zeta function, and have a Dirichlet series expansion and a functionalequation, but the ones known to fail the Riemann hypothesis do not have an Euler product and are not directlyrelated to automorphic representations.

• The numerical verification that many zeros lie on the line seems at first sight to be strong evidence for it.However analytic number theory has had many conjectures supported by large amounts of numerical evidencethat turn out to be false. See Skewes number for a notorious example, where the first exception to a plausibleconjecture related to the Riemann hypothesis probably occurs around 10316; a counterexample to the Riemannhypothesis with imaginary part this size would be far beyond anything that can currently be computed. Theproblem is that the behavior is often influenced by very slowly increasing functions such as log log T, that tend toinfinity, but do so so slowly that this cannot be detected by computation. Such functions occur in the theory of thezeta function controlling the behavior of its zeros; for example the function S(T) above has average size around(log log T)1/2 . As S(T) jumps by at least 2 at any counterexample to the Riemann hypothesis, one might expectany counterexamples to the Riemann hypothesis to start appearing only when S(T) becomes large. It is nevermuch more than 3 as far as it has been calculated, but is known to be unbounded, suggesting that calculations maynot have yet reached the region of typical behavior of the zeta function.

• Denjoy's probabilistic argument for the Riemann hypothesis (Edwards 1974) is based on the observation that Ifμ(x) is a random sequence of "1"s and "−1"s then, for every ε > 0, the partial sums

(the values of which are positions in a simple random walk) satisfy the bound

with probability 1. The Riemann hypothesis is equivalent to this bound for the Möbius function μ and theMertens function M derived in the same way from it. In other words, the Riemann hypothesis is in some senseequivalent to saying that μ(x) behaves like a random sequence of coin tosses. When μ(x) is non-zero its signgives the parity of the number of prime factors of x, so informally the Riemann hypothesis says that the parityof the number of prime factors of an integer behaves randomly. Such probabilistic arguments in number theoryoften give the right answer, but tend to be very hard to make rigorous, and occasionally give the wrong answerfor some results, such as Maier's theorem.

• The calculations in Odlyzko (1987) show that the zeros of the zeta function behave very much like theeigenvalues of a random Hermitian matrix, suggesting that they are the eigenvalues of some self-adjoint operator,which would imply the Riemann hypothesis. However all attempts to find such an operator have failed.

• There are several theorems, such as Goldbach's conjecture for sufficiently large odd numbers, that were firstproved using the generalized Riemann hypothesis, and later shown to be true unconditionally. This could beconsidered as weak evidence for the generalized Riemann hypothesis, as several of its "predictions" turned out tobe true.

• Lehmer's phenomenon (Lehmer 1956) where two zeros are sometimes very close is sometimes given as a reasonto disbelieve in the Riemann hypothesis. However one would expect this to happen occasionally just by chanceeven if the Riemann hypothesis were true, and Odlyzko's calculations suggest that nearby pairs of zeros occur justas often as predicted by Montgomery's conjecture.

Riemann hypothesis 38

• Patterson (1988) suggests that the most compelling reason for the Riemann hypothesis for most mathematicians isthe hope that primes are distributed as regularly as possible.

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• Titchmarsh, Edward Charles (1935), "The Zeros of the Riemann Zeta-Function", Proceedings of the RoyalSociety of London. Series A, Mathematical and Physical Sciences (The Royal Society) 151 (873): 234–255,doi:10.1098/rspa.1935.0146, JSTOR 96545

• Titchmarsh, Edward Charles (1936), "The Zeros of the Riemann Zeta-Function", Proceedings of the RoyalSociety of London. Series A, Mathematical and Physical Sciences (The Royal Society) 157 (891): 261–263,doi:10.1098/rspa.1936.0192, JSTOR 96692

• Titchmarsh, Edward Charles (1986), The theory of the Riemann zeta-function (2nd ed.), The Clarendon PressOxford University Press, ISBN 978-0-19-853369-6, MR882550

• Turán, Paul (1948), "On some approximative Dirichlet-polynomials in the theory of the zeta-function ofRiemann", Danske Vid. Selsk. Mat.-Fys. Medd. 24 (17): 36, MR0027305 Reprinted in (Borwein et al. 2008).

• Turing, Alan M. (1953), "Some calculations of the Riemann zeta-function", Proceedings of the LondonMathematical Society. Third Series 3: 99–117, doi:10.1112/plms/s3-3.1.99, MR0055785

Riemann hypothesis 42

• de la Vallée-Poussin, Ch.J. (1896), "Recherches analytiques sur la théorie des nombers premiers", Ann. Soc. Sci.Bruxelles 20: 183–256

• de la Vallée-Poussin, Ch.J. (1899–1900), "Sur la fonction ζ(s) de Riemann et la nombre des nombres premiersinférieurs à une limite donnée", Mem. Couronnes Acad. Sci. Belg. 59 (1) Reprinted in (Borwein et al. 2008).

• Weil, André (1948), Sur les courbes algébriques et les variétés qui s'en déduisent, Actualités Sci. Ind., no. 1041 =Publ. Inst. Math. Univ. Strasbourg 7 (1945), Hermann et Cie., Paris, MR0027151

• Weil, André (1949), "Numbers of solutions of equations in finite fields", Bulletin of the American MathematicalSociety 55 (5): 497–508, doi:10.1090/S0002-9904-1949-09219-4, MR0029393 Reprinted in OeuvresScientifiques/Collected Papers by Andre Weil ISBN 0-387-90330-5

• Weinberger, Peter J. (1973), "On Euclidean rings of algebraic integers", Analytic number theory ( St. Louis Univ.,1972), Proc. Sympos. Pure Math., 24, Providence, R.I.: Amer. Math. Soc., pp. 321–332, MR0337902

• Wiles, Andrew (2000), "Twenty years of number theory", Mathematics: frontiers and perspectives, Providence,R.I.: American Mathematical Society, pp. 329–342, ISBN 978-0-8218-2697-3, MR1754786

• Zagier, Don (1977), "The first 50 million prime numbers" [25] (PDF), Math. Intelligencer (Springer) 0: 7–19,doi:10.1007/BF03039306, MR643810

• Zagier, Don (1981), "Eisenstein series and the Riemann zeta function", Automorphic forms, representation theoryand arithmetic (Bombay, 1979), Tata Inst. Fund. Res. Studies in Math., 10, Tata Inst. Fundamental Res., Bombay,pp. 275–301, MR633666

External links• American institute of mathematics, Riemann hypothesis [26]

• Apostol, Tom, Where are the zeros of zeta of s? [27] Poem about the Riemann hypothesis, sung [28] by JohnDerbyshire.

• Borwein, Peter (PDF), The Riemann Hypothesis [29] (Slides for a lecture)• Conrad, K. (2010), Consequences of the Riemann hypothesis [30]

• Conrey, J. Brian; Farmer, David W, Equivalences to the Riemann hypothesis [31]

• Gourdon, Xavier; Sebah, Pascal (2004), Computation of zeros of the Zeta function [32] (Reviews the GUEhypothesis, provides an extensive bibliography as well).

• Odlyzko, Andrew, Home page [33] including papers on the zeros of the zeta function [34] and tables of the zeros ofthe zeta function [35]

• Odlyzko, Andrew (2002) (PDF), Zeros of the Riemann zeta function: Conjectures and computations [36] Slides ofa talk

• Pegg, Ed (2004), Ten Trillion Zeta Zeros [37], Math Games website. A discussion of Xavier Gourdon's calculationof the first ten trillion non-trivial zeros

• Pugh, Glen, Java applet for plotting Z(t) [38]

• Rubinstein, Michael, algorithm for generating the zeros [39].• du Sautoy, Marcus (2006), Prime Numbers Get Hitched [40], Seed Magazine [41]

• Stein, William A., What is Riemann's hypothesis [42]

• de Vries, Andreas (2004), The Graph of the Riemann Zeta function ζ(s) [43], a simple animated Java applet.• Watkins, Matthew R. (2007-07-18), Proposed proofs of the Riemann Hypothesis [44]

• Zetagrid [45] (2002) A distributed computing project that attempted to disprove Riemann's hypothesis; closed inNovember 2005

Riemann hypothesis 43

References[1] http:/ / arxiv. org/ find/ grp_math/ 1/ AND+ ti:+ AND+ Riemann+ hypothesis+ subj:+ AND+ General+ mathematics/ 0/ 1/ 0/ all/ 0/ 1[2] http:/ / en. wikipedia. org/ wiki/ Oeis%3Aa114856[3] http:/ / www. claymath. org/ millennium/ Riemann_Hypothesis/ riemann. pdf[4] http:/ / www. digizeitschriften. de/ resolveppn/ GDZPPN002206781[5] http:/ / www. ams. org/ notices/ 200303/ fea-conrey-web. pdf[6] http:/ / www. numdam. org/ item?id=PMIHES_1974__43__273_0[7] http:/ / www. numdam. org/ item?id=PMIHES_1980__52__137_0[8] http:/ / www. mathematik. uni-bielefeld. de/ documenta/ xvol-icm/ 00/ Deninger. MAN. html[9] http:/ / www. ams. org/ notices/ 200902/ rtx090200212p. pdf[10] http:/ / numbers. computation. free. fr/ Constants/ Miscellaneous/ zetazeros1e13-1e24. pdf[11] http:/ / www. numdam. org/ item?id=BSMF_1896__24__199_1[12] http:/ / gallica. bnf. fr/ ark:/ 12148/ bpt6k3111d. image. f1014. langEN[13] http:/ / www. jstor. org/ stable/ 2003098[14] http:/ / eom. springer. de/ Z/ z099260. htm[15] http:/ / matwbn. icm. edu. pl/ tresc. php?wyd=6& tom=50& jez=[16] http:/ / www. trnicely. net/ gaps/ gaps. html[17] http:/ / gdz. sub. uni-goettingen. de/ no_cache/ dms/ load/ img/ ?IDDOC=262633[18] http:/ / www. numdam. org/ item?id=JTNB_1990__2_1_119_0[19] http:/ / www. dtc. umn. edu/ ~odlyzko/ unpublished/ zeta. 10to20. 1992. pdf[20] http:/ / www. dtc. umn. edu/ ~odlyzko/ unpublished/ zeta. 10to21. pdf[21] http:/ / www. maths. tcd. ie/ pub/ HistMath/ People/ Riemann/ Zeta/[22] http:/ / www. claymath. org/ millennium/ Riemann_Hypothesis/ 1859_manuscript/[23] http:/ / www. claymath. org/ millennium/ Riemann_Hypothesis/ Sarnak_RH. pdf[24] http:/ / modular. math. washington. edu/ edu/ 2007/ simuw07/ notes/ rh. pdf[25] http:/ / modular. math. washington. edu/ edu/ 2007/ simuw07/ misc/ zagier-the_first_50_million_prime_numbers. pdf[26] http:/ / www. aimath. org/ WWN/ rh/[27] http:/ / www. math. wisc. edu/ ~robbin/ funnysongs. html#Zeta[28] http:/ / www. olimu. com/ RIEMANN/ Song. htm[29] http:/ / oldweb. cecm. sfu. ca/ ~pborwein/ COURSE/ MATH08/ LECTURE. pdf[30] http:/ / mathoverflow. net/ questions/ 17232[31] http:/ / aimath. org/ pl/ rhequivalences[32] http:/ / numbers. computation. free. fr/ Constants/ Miscellaneous/ zetazeroscompute. html[33] http:/ / www. dtc. umn. edu/ ~odlyzko/[34] http:/ / www. dtc. umn. edu/ ~odlyzko/ doc/ zeta. html[35] http:/ / www. dtc. umn. edu/ ~odlyzko/ zeta_tables/ index. html[36] http:/ / www. dtc. umn. edu/ ~odlyzko/ talks/ riemann-conjectures. pdf[37] http:/ / www. maa. org/ editorial/ mathgames/ mathgames_10_18_04. html[38] http:/ / web. viu. ca/ pughg/ RiemannZeta/ RiemannZetaLong. html[39] http:/ / pmmac03. math. uwaterloo. ca/ ~mrubinst/ l_function_public/ L. html[40] http:/ / www. seedmagazine. com/ news/ 2006/ 03/ prime_numbers_get_hitched. php[41] http:/ / www. seedmagazine. com[42] http:/ / modular. math. washington. edu/ edu/ 2007/ simuw07/ index. html[43] http:/ / math-it. org/ Mathematik/ Riemann/ RiemannApplet. html[44] http:/ / secamlocal. ex. ac. uk/ ~mwatkins/ zeta/ RHproofs. htm[45] http:/ / www. zetagrid. net/

Navier–Stokes existence and smoothness 44

Navier–Stokes existence and smoothness

Millennium Prize Problems

P versus NP problem

Hodge conjecture

Poincaré conjecture (solution)

Riemann hypothesis

Yang–Mills existence and mass gap

Navier–Stokes existence and smoothness

Birch and Swinnerton-Dyer conjecture

The Navier–Stokes equations are one of the pillars of fluid mechanics. These equations describe the motion of afluid (that is, a liquid or a gas) in space. Solutions to the Navier–Stokes equations are used in many practicalapplications. However, theoretical understanding of the solutions to these equations is incomplete. In particular,solutions of the Navier–Stokes equations often include turbulence, which remains one of the greatest unsolvedproblems in physics despite its immense importance in science and engineering.Even much more basic properties of the solutions to Navier–Stokes have never been proven. For thethree-dimensional system of equations, and given some initial conditions, mathematicians have not yet proved thatsmooth solutions always exist, or that if they do exist they have bounded kinetic energy. This is called theNavier–Stokes existence and smoothness problem.Since understanding the Navier–Stokes equations is considered to be the first step for understanding the elusivephenomenon of turbulence, the Clay Mathematics Institute offered in May 2000 a US$1,000,000 prize, not towhomever constructs a theory of turbulence but (more modestly) to the first person providing a hint on thephenomenon of turbulence. In that spirit of ideas, the Clay Institute set a concrete mathematical problem:[1]

Prove or give a counter-example of the following statement:In three space dimensions and time, given an initial velocity field, there exists a vector velocity and a scalarpressure field, which are both smooth and globally defined, that solve the Navier–Stokes equations.

The Navier–Stokes equationsIn mathematics, the Navier–Stokes equations are a system of nonlinear partial differential equations for abstractvector fields of any size. In physics and engineering, they are a system of equations that models the motion of liquidsor non-rarefied gases using continuum mechanics. The equations are a statement of Newton's second law, with theforces modeled according to those in a viscous Newtonian fluid—as the sum of contributions by pressure, viscousstress and an external body force. Since the setting of the problem proposed by the Clay Mathematics Institute is inthree dimensions, for an incompressible and homogeneous fluid, we will consider only that case.

Let be a 3-dimensional vector, the velocity of the fluid, and let be the pressure of the fluid.[2] TheNavier–Stokes equations are:

where is the kinematic viscosity, the external force, is the gradient operator and is theLaplacian operator, which is also denoted by . Note that this is a vector equation, i.e. it has three scalarequations. If we write down the coordinates of the velocity and the external force

Navier–Stokes existence and smoothness 45

then for each we have the corresponding scalar Navier–Stokes equation:

The unknowns are the velocity and the pressure . Since in three dimensions we have threeequations and four unknowns (three scalar velocities and the pressure), we need a supplementary equation. Thisextra equation is the continuity equation describing the incompressibility of the fluid:

Due to this last property, the solutions for the Navier–Stokes equations are searched in the set of "divergence-free"functions. For this flow of a homogeneous medium, density and viscosity are constants.We can eliminate the pressure p by taking an operator rot (alternative notation curl) of both sides of theNavier–Stokes equations. In this case the Navier–Stokes equations reduce to the Vorticity transport equations. Intwo dimensions (2D), these equations are well known [6, p. 321]. In three dimensions (3D), it is known for a longtime that Vorticity transport equations have additional terms [6, p. 294]. However, why 1D, 2D and 3DNavier–Stokes equations in the vector form are identical? In that case, probably, the vorticity transport equations inthe vector form must be identical too.

Two settings: unbounded and periodic spaceThere are two different settings for the one-million-dollar-prize Navier–Stokes existence and smoothness problem.The original problem is in the whole space , which needs extra conditions on the growth behavior of the initialcondition and the solutions. In order to rule out the problems at infinity, the Navier–Stokes equations can be set in aperiodic framework, which implies that we are no longer working on the whole space but in the 3-dimensionaltorus . We will treat each case separately.

Statement of the problem in the whole space

Hypotheses and growth conditions

The initial condition is assumed to be a smooth and divergence-free function (see smooth function) suchthat, for every multi-index (see multi-index notation) and any , there exists a constant

(i.e. this "constant" depends on and K) such that

for all

The external force is assumed to be a smooth function as well, and satisfies a very analogous inequality(now the multi-index includes time derivatives as well):

for all

For physically reasonable conditions, the type of solutions expected are smooth functions that do not grow large as. More precisely, the following assumptions are made:

1.

2. There exists a constant such that for all

Condition 1 implies that the functions are smooth and globally defined and condition 2 means that the kinetic energyof the solution is globally bounded.

Navier–Stokes existence and smoothness 46

The million-dollar-prize conjectures in the whole space

(A) Existence and smoothness of the Navier–Stokes solutions in

Let . For any initial condition satisfying the above hypotheses there exist smooth and globallydefined solutions to the Navier–Stokes equations, i.e. there is a velocity vector and a pressure satisfying conditions 1 and 2 above.(B) Breakdown of the Navier–Stokes solutions in

There exists an initial condition and an external force such that there exists no solutions and satisfying conditions 1 and 2 above.

Statement of the periodic problem

HypothesesThe functions we seek now are periodic in the space variables of period 1. More precisely, let be the unitaryvector in the j- direction:

Then is periodic in the space variables if for any we have that

Notice that we are considering the coordinates mod 1. This allows us to work not on the whole space but on thequotient space , which turns out to be the 3-dimensional torus

We can now state the hypotheses properly. The initial condition is assumed to be a smooth anddivergence-free function and the external force is assumed to be a smooth function as well. The type ofsolutions that are physically relevant are those who satisfy these conditions:3.

4. There exists a constant such that for all

Just as in the previous case, condition 3 implies that the functions are smooth and globally defined and condition 4means that the kinetic energy of the solution is globally bounded.

The periodic million-dollar-prize theorems

(C) Existence and smoothness of the Navier–Stokes solutions in

Let . For any initial condition satisfying the above hypotheses there exist smooth and globallydefined solutions to the Navier–Stokes equations, i.e. there is a velocity vector and a pressure satisfying conditions 3 and 4 above.(D) Breakdown of the Navier–Stokes solutions in

There exists an initial condition and an external force such that there exists no solutions and satisfying conditions 3 and 4 above.

Navier–Stokes existence and smoothness 47

Partial results1. The Navier–Stokes problem in two dimensions has already been solved positively since the 1960s: there exist

smooth and globally defined solutions.[3]

2. If the initial velocity is sufficiently small then the statement is true: there are smooth and globallydefined solutions to the Navier–Stokes equations.[1]

3. Given an initial velocity there exists a finite time T, depending on such that the Navier–Stokesequations on have smooth solutions and . It is not known if the solutions existbeyond that "blowup time" T.[1]

4. The mathematician Jean Leray in 1934 proved the existence of so called weak solutions to the Navier–Stokesequations, satisfying the equations in mean value, not pointwise.[4]

Notes[1] Official statement of the problem (http:/ / www. claymath. org/ millennium/ Navier-Stokes_Equations/ navierstokes. pdf), Clay Mathematics

Institute.

[2] More precisely, is the pressure divided by the fluid density, and the density is constant for this incompressible and homogeneous

fluid.[3] Ladyzhenskaya, O. (1969), The Mathematical Theory of Viscous Incompressible Flows (2nd ed.), New York: Gordon and Breach.[4] Leray, J. (1934), "Sur le mouvement d'un liquide visqueux emplissant l'espace", Acta Mathematica 63: 193–248, doi:10.1007/BF02547354

References

External links• The Clay Mathematics Institute's Navier–Stokes equation prize (http:/ / www. claymath. org/ millennium/

Navier-Stokes_Equations/ )• Why global regularity for Navier–Stokes is hard (http:/ / terrytao. wordpress. com/ 2007/ 03/ 18/

why-global-regularity-for-navier-stokes-is-hard) — Possible routes to resolution are scrutinized by Terence Tao.• Fuzzy Fluid Mechanics (http:/ / sgrajeev. com/ fuzzy-fluids/ )• Navier–Stokes existence and smoothness (Millennium Prize Problem) (http:/ / vimeo. com/ 18185364/ ) A lecture

on the problem by Luis Caffarelli.

Birch and Swinnerton-Dyer conjecture 48

Birch and Swinnerton-Dyer conjecture

Millennium Prize Problems

P versus NP problem

Hodge conjecture

Poincaré conjecture (solution)

Riemann hypothesis

Yang–Mills existence and mass gap

Navier–Stokes existence and smoothness

Birch and Swinnerton-Dyer conjecture

In mathematics, the Birch and Swinnerton-Dyer conjecture is an open problem in the field of number theory. Itsstatus as one of the most challenging mathematical questions has become widely recognized; the conjecture waschosen as one of the seven Millennium Prize Problems listed by the Clay Mathematics Institute, which has offered a$1,000,000 prize for the first correct proof. As of 2010, only special cases of the conjecture have been provedcorrect.The conjecture relates arithmetic data associated to an elliptic curve E over a number field K to the behaviour of theHasse-Weil L-function L(E, s) of E at s = 1. More specifically, it is conjectured that the rank of the abelian groupE(K) of points of E is the order of the zero of L(E, s) at s = 1, and the first non-zero coefficient in the Taylorexpansion of L(E, s) at s = 1 is given by more refined arithmetic data attached to E over K.[1]

BackgroundIn 1922 Louis Mordell proved Mordell's theorem: the group of rational points on an elliptic curve has a finite basis.This means that for any elliptic curve there is a finite sub-set of the rational points on the curve, from which allfurther rational points may be generated.If the number of rational points on a curve is infinite then some point in a finite basis must have infinite order. Thenumber of independent basis points with infinite order is called the rank of the curve, and is an important invariantproperty of an elliptic curve.If the rank of an elliptic curve is 0, then the curve has only a finite number of rational points. On the other hand, ifthe rank of the curve is greater than 0, then the curve has an infinite number of rational points.Although Mordell's theorem shows that the rank of an elliptic curve is always finite, it does not give an effectivemethod for calculating the rank of every curve. The rank of certain elliptic curves can be calculated using numericalmethods but (in the current state of knowledge) these cannot be generalised to handle all curves.An L-function L(E, s) can be defined for an elliptic curve E by constructing an Euler product from the number ofpoints on the curve modulo each prime p. This L-function is analogous to the Riemann zeta function and theDirichlet L-series that is defined for a binary quadratic form. It is a special case of a Hasse-Weil L-function.The natural definition of L(E, s) only converges for values of s in the complex plane with Re(s) > 3/2. Helmut Hasseconjectured that L(E, s) could be extended by analytic continuation to the whole complex plane. This conjecture wasfirst proved by Max Deuring for elliptic curves with complex multiplication. It was subsequently shown to be truefor all elliptic curves over Q, as a consequence of the modularity theorem.Finding rational points on a general elliptic curve is a difficult problem. Finding the points on an elliptic curvemodulo a given prime p is conceptually straightforward, as there are only a finite number of possibilities to check.However, for large primes it is computationally intensive.

Birch and Swinnerton-Dyer conjecture 49

HistoryIn the early 1960s Peter Swinnerton-Dyer used the EDSAC computer at the University of Cambridge ComputerLaboratory to calculate the number of points modulo p (denoted by Np) for a large number of primes p on ellipticcurves whose rank was known. From these numerical results Bryan Birch and Swinnerton-Dyer conjectured that Npfor a curve E with rank r obeys an asymptotic law

A plot of for the curve y2 = x3 − 5x as X varies over the first 100000 primes.

The X-axis is log(log(X)) and Y-axis is in a logarithmic scale so the conjecture predictsthat the data should form a line of slope equal to the rank of the curve, which is 1 in this

case. For comparison, a line of slope 1 is drawn in red on the graph.

where C is a constant.Initially this was based on somewhat tenuous trends in graphical plots; which induced a measure of skepticism in J.W. S. Cassels (Birch's Ph.D. advisor). Over time the numerical evidence stacked up.This in turn led them to make a general conjecture about the behaviour of a curve's L-function L(E, s) at s = 1,namely that it would have a zero of order r at this point. This was a far-sighted conjecture for the time, given that theanalytic continuation of L(E, s) there was only established for curves with complex multiplication, which were alsothe main source of numerical examples. (NB that the reciprocal of the L-function is from some points of view a morenatural object of study; on occasion this means that one should consider poles rather than zeroes.)The conjecture was subsequently extended to include the prediction of the precise leading Taylor coefficient of theL-function at s = 1. It is conjecturally given by

where the quantities on the right hand side are invariants of the curve, studied by Cassels, Tate, Shafarevich andothers: these include the order of the torsion group, the order of the Tate-Shafarevich group, and the canonicalheights of a basis of rational points.[1]

Birch and Swinnerton-Dyer conjecture 50

Current statusThe Birch and Swinnerton-Dyer conjecture has been proved only in special cases :1. In 1976, John Coates and Andrew Wiles proved that if E is a curve over a number field F with complex

multiplication by an imaginary quadratic field K of class number 1, F = K or Q, and L(E, 1) is not 0 then E(F) is afinite group.[2] This was extended to the case where F is any finite abelian extension of K by NicoleArthaud-Kuhman.[3]

2. In 1983, Benedict Gross and Don Zagier showed that if a modular elliptic curve has a first-order zero at s = 1 thenit has a rational point of infinite order;[4] see Gross–Zagier theorem.

3. In 1990, Victor Kolyvagin showed that a modular elliptic curve E for which L(E,1) is not zero has rank 0, and amodular elliptic curve E for which L(E,1) has a first-order zero at s = 1 has rank 1.

4. In 1991, Karl Rubin showed that for elliptic curves defined over an imaginary quadratic field K with complexmultiplication by K, if the L-series of the elliptic curve was not zero at s=1, then the p-part of theTate–Shafarevich group had the order predicted by the Birch and Swinnerton-Dyer conjecture, for all primes p >7.[5]

5. In 2001, Christophe Breuil, Brian Conrad, Fred Diamond and Richard Taylor, extending work of Wiles, provedthat all elliptic curves defined over the rational numbers are modular (the Taniyama-Shimura theorem), whichextends results 2 and 3 to all elliptic curves over the rationals, and shows that the L-functions of all elliptic curvesover Q are defined at s = 1.[6]

6. In 2010, Manjul Bhargava and Arul Shankar announced a proof that the average rank of the Mordell–Weil groupof an elliptic curve over Q is bounded above by 7/6.[7] Combining this with the announced proof of the mainconjecture of Iwasawa theory for GL(2) by Chris Skinner and Éric Urban,[8] they conclude that a positiveproportion of elliptic curves over Q have analytic rank zero, and hence, by Kolyvagin's result, satisfy the Birchand Swinnerton-Dyer conjecture.

Nothing has been proved for curves with rank greater than 1, although there is extensive numerical evidence for thetruth of the conjecture.

Clay Mathematics Institute PrizeThe Birch and Swinnerton-Dyer conjecture is one of the seven Millennium Problems selected by the ClayMathematics Institute, which is offering a prize of $1 million for the first proof or disproof of the wholeconjecture.[9]

Notes[1] Wiles 2006[2] Coates, J.; Wiles, A. (1977). "On the conjecture of Birch and Swinnerton-Dyer". Inventiones Mathematicae 39 (3): 223–251.

doi:10.1007/BF01402975.[3] Arthaud, Nicole (1978). "On Birch and Swinnerton-Dyer's conjecture for elliptic curves with complex multiplication". Compositio

Mathematica 37 (2): 209–232. MR504632.[4] Gross, Benedict H.; Zagier, Don B. (1986). "Heegner points and derivatives of L-series". Inventiones Mathematicae 84 (2): 225–320.

doi:10.1007/BF01388809. MR0833192.[5] Rubin, Karl (1991). "The 'main conjectures' of Iwasawa theory for imaginary quadratic fields". Inventiones Mathematicae 103 (1): 25–68.

doi:10.1007/BF01239508.[6] Breuil, Christophe; Conrad, Brian; Diamond, Fred; Taylor, Richard (2001). "On the Modularity of Elliptic Curves over Q: Wild 3-Adic

Exercises". Journal of the American Mathematical Society 14 (4): 843–939. doi:10.1090/S0894-0347-01-00370-8.[7] Bhargava, Manjul; Shankar, Arul (2010). "Ternary cubic forms having bounded invariants, and the existence of a positive proportion of

elliptic curves having rank 0". Preprint. arXiv:1007.0052.[8] Skinner, Chris; Urban, Éric (2010). "The Iwasawa main conjectures for GL2" (http:/ / www. math. columbia. edu/ ~urban/ eurp/ MC. pdf). In

preparation. .[9] Birch and Swinnerton-Dyer Conjecture (http:/ / www. claymath. org/ millennium/ Birch_and_Swinnerton-Dyer_Conjecture/ ) at Clay

Mathematics Institute

Birch and Swinnerton-Dyer conjecture 51

References• Wiles, Andrew (2006), "The Birch and Swinnerton-Dyer conjecture" (http:/ / www. claymath. org/ millennium/

Birch_and_Swinnerton-Dyer_Conjecture/ birchswin. pdf), in Carlson, James; Jaffe, Arthur; Wiles, Andrew, TheMillennium prize problems, American Mathematical Society, pp. 31–44, ISBN 978-0-821-83679-8

External links• Weisstein, Eric W., " Swinnerton-Dyer Conjecture (http:/ / mathworld. wolfram. com/

Swinnerton-DyerConjecture. html)" from MathWorld.• Birch and Swinnerton-Dyer Conjecture (http:/ / planetmath. org/ ?op=getobj& amp;from=objects& amp;id=4561)

on PlanetMath• The Birch and Swinnerton-Dyer Conjecture (http:/ / sums. mcgill. ca/ delta-epsilon/ mag/ 0610/ mmm061024.

pdf): An Interview with Professor Henri Darmon by Agnes F. Beaudry

Yang–Mills existence and mass gapThe Clay Mathematics Institute has offered a prize of US$1,000,000 to a person solving any one of the MillenniumPrize Problems of modern mathematics. One of these seven problems is phrased as follows:

Yang–Mills Existence and Mass Gap. Prove that for any compact simple gauge group G, a non-trivialquantum Yang–Mills theory exists on R4 and has a mass gap Δ > 0. Existence includes establishing axiomaticproperties at least as strong as those cited in [45, 35].

In this statement, Yang–Mills theory is the (non-Abelian) quantum field theory underlying the Standard Model ofparticle physics; R4 is space-time; the mass gap Δ is the mass of the least massive particle predicted by the theory.Therefore, the winner must first prove that Yang–Mills theory exists and that it satisfies the standard of rigor thatcharacterizes contemporary mathematical physics, in particular constructive quantum field theory, which isreferenced in the papers 45 and 35 cited in the official problem description by Jaffe and Witten. The winner mustthen prove that the mass of the least massive particle of the force field predicted by the theory is strictly positive. Forexample, in the case of G=SU(3) - the strong nuclear interaction - the winner must prove that glueballs have a lowermass bound, and thus cannot be arbitrarily light.

Millennium Prize Problems

P versus NP problem

Hodge conjecture

Poincaré conjecture (solution)

Riemann hypothesis

Yang–Mills existence and mass gap

Navier–Stokes existence and smoothness

Birch and Swinnerton-Dyer conjecture

Yang–Mills existence and mass gap 52

Background

“[...] one does not yet have a mathematically complete example of a quantum gauge theory in four-dimensional space-time, nor even a precisedefinition of quantum gauge theory in four dimensions. Will this change in the 21st century? We hope so! ”

—From the Clay Institute's official problem description by Arthur Jaffe and Edward Witten.

Most known and nontrivial (i.e. interacting) quantum field theories in 4 dimensions are effective field theories with acutoff scale. Since the beta-function is positive for most models, it appears that most such models have a Landaupole as it is not at all clear whether or not they have nontrivial UV fixed points. This means that if such a QFT iswell-defined at all scales, as it has to be to satisfy the axioms of axiomatic quantum field theory, it would have to betrivial (i.e. a free field theory).Quantum Yang-Mills theory with a non-abelian gauge group and no quarks is an exception, because asymptoticfreedom characterizes this theory, meaning that it has a trivial UV fixed point. Hence it is the simplest nontrivialconstructive QFT in 4 dimensions. (QCD is a more complicated theory because it involves quarks.)It has already been well proven—at least at the level of rigor of theoretical physics but not that of mathematicalphysics—that the quantum Yang–Mills theory for a non-abelian Lie group exhibits a property known asconfinement. This property is covered in more detail in the relevant QCD articles (QCD, color confinement, latticegauge theory, etc.), although not at the level of rigor of mathematical physics. A consequence of this property is thatbeyond a certain scale, known as the QCD scale (more properly, the confinement scale, as this theory is devoid ofquarks), the color charges are connected by chromodynamic flux tubes leading to a linear potential between thecharges. Hence free color charge and free gluons cannot exist. In the absence of confinement, we would expect to seemassless gluons, but since they are confined, all we see are color-neutral bound states of gluons, called glueballs. Ifglueballs exist, they are massive, which is why we expect a mass gap.Results from lattice gauge theory have convinced many that quantum Yang–Mills theory for a non-abelian Lie groupmodel exhibits confinement—as indicated, for example, by an area law for the falloff of the vacuum expectationvalue (VEV) of a Wilson loop. However, these methods and results are not mathematically rigorous.

References• Arthur Jaffe and Edward Witten "Quantum Yang-Mills theory. [1]" Official problem description.

External links• The Millennium Prize Problems: Yang–Mills and Mass Gap [2]

References[1] http:/ / www. claymath. org/ millennium/ Yang-Mills_Theory/ yangmills. pdf[2] http:/ / www. claymath. org/ millennium/ Yang-Mills_Theory/

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Hodge conjecture  Source: http://en.wikipedia.org/w/index.php?oldid=419614816  Contributors: Baccyak4H, BenTheMen, Bender235, Benzh, BerndGehrmann, Bihco, Changbao, CharlesMatthews, ChicXulub, ComplexZeta, Cícero, DanielDeibler, David Haslam, Dfrg.msc, Drbreznjev, Eoladis, Gareth Jones, Giftlite, Gtrmp, Hofingerandi, Ilmari Karonen, Jakob.scholbach,Jtwdog, KSmrq, LJosil, Master Bratac, Michael Hardy, Michael Slone, Nzseries1, Ozob, Pruneau, Silvonen, Snowdog, Spireguy, Steven1234321, Suruena, Sławomir Biały, The Anome, Tong,UkPaolo, Willking1979, ZX81, 46 anonymous edits

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Birch and Swinnerton-Dyer conjecture  Source: http://en.wikipedia.org/w/index.php?oldid=428916980  Contributors: Ampassag, Bender235, Borat fan, CBM, CRGreathouse, CharlesMatthews, CryptoDerk, David Eppstein, David Haslam, DiceDiceBaby, Epsilon17, Gandalf61, Gauge, Giftlite, Hesam7, Hofingerandi, J.delanoy, Jaraalbe, Jpbowen, Jtwdog, Lenthe, Linas,Lowellian, MathMartin, Maximus Rex, Michael Hardy, Oleg Yunakov, Phe, Qutezuce, R.e.b., Rajpaj, Raulshc, Rhythm, RobHar, Robertwb, Saga City, Samwb123, Squeezy, Suruena, Tbsmith,Thewhyman, Timothy Clemans, Timwi, UkPaolo, Waltpohl, Zoicon5, 40 anonymous edits

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