Bernoulli number 1

Bernoulli numberIn mathematics, the Bernoulli numbers Bn are a sequence of rational numbers with deep connections to numbertheory. The values of the first few Bernoulli numbers are

B0 = 1, B1 = ±1⁄2, B2 = 1⁄6, B3 = 0, B4 = −1⁄30, B5 = 0, B6 = 1⁄42, B7 = 0, B8 = −1⁄30.If the convention B1=−1⁄2 is used, this sequence is also known as the first Bernoulli numbers (A027641 / A027642in OEIS); with the convention B1=+1⁄2 is known as the second Bernoulli numbers (A164555 / A027642 in OEIS).Except for this one difference, the first and second Bernoulli numbers agree. Since Bn=0 for all odd n>1, and manyformulas only involve even-index Bernoulli numbers, some authors write Bn instead of B2n.The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, informulas for the sum of powers of the first positive integers, in the Euler–Maclaurin formula, and in expressions forcertain values of the Riemann zeta function.The Bernoulli numbers were discovered around the same time by the Swiss mathematician Jakob Bernoulli, afterwhom they are named, and independently by Japanese mathematician Seki Kōwa. Seki's discovery wasposthumously published in 1712[1][2] in his work Katsuyo Sampo; Bernoulli's, also posthumously, in his ArsConjectandi of 1713. Ada Lovelace's note G on the analytical engine from 1842 describes an algorithm forgenerating Bernoulli numbers with Babbage's machine.[3] As a result, the Bernoulli numbers have the distinction ofbeing the subject of the first computer program.

Sum of powersBernoulli numbers feature prominently in the closed form expression of the sum of the m-th powers of the first npositive integers. For m, n ≥ 0 define

This expression can always be rewritten as a polynomial in n of degree m + 1. The coefficients of these polynomialsare related to the Bernoulli numbers by Bernoulli's formula:

where the convention B1 = +1/2 is used. ( denotes the binomial coefficient, m+1 choose k.)For example, taking m to be 1 gives the triangular numbers 0, 1, 3, 6, ... (sequence A000217 in OEIS).

Taking m to be 2 gives the square pyramidal numbers 0, 1, 5, 14, ... (sequence A000330 in OEIS).

Some authors use the convention B1 = −1/2 and state Bernoulli's formula in this way:

Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who also found remarkable waysto calculate sum of powers.Faulhaber's formula was generalized by V. Guo and J. Zeng to a q-analog (Guo & Zeng 2005).

Bernoulli number 2

DefinitionsMany characterizations of the Bernoulli numbers have been found in the last 300 years, and each could be used tointroduce these numbers. Here only four of the most useful ones are mentioned:•• a recursive equation,•• an explicit formula,•• a generating function,•• an algorithmic description.For the proof of the equivalence of the four approaches the reader is referred to mathematical expositions like(Ireland & Rosen 1990) or (Conway & Guy 1996).Unfortunately in the literature the definition is given in two variants: Despite the fact that Bernoulli defined B1 = 1/2(now known as "second Bernoulli numbers"), some authors set B1 = −1/2 ("first Bernoulli numbers"). In order toprevent potential confusions both variants will be described here, side by side. Because these two definitions can betransformed simply by into the other, some formulae have this alternatingly (-1)n-term and othersnot depending on the context, but it is not possible to decide in favor of one of these definitions to be the correct orappropriate or natural one (for the abstract Bernoulli numbers).

Recursive definitionThe recursive equation is best introduced in a slightly more general form

This defines polynomials Bm in the variable n known as the Bernoulli polynomials. The recursion can also be viewedas defining rational numbers Bm(n) for all integers n ≥ 0, m ≥ 0. The expression 00 has to be interpreted as 1. Thefirst and second Bernoulli numbers now follow by setting n = 0 (resulting in B1=−1⁄2, "first Bernoulli numbers")respectively n = 1 (resulting in B1=+1⁄2, "second Bernoulli numbers").

Here the expression [m = 0] has the value 1 if m = 0 and 0 otherwise (Iverson bracket). Whenever a confusionbetween the two kinds of definitions might arise it can be avoided by referring to the more general definition and byreintroducing the erased parameter: writing Bm(0) in the first case and Bm(1) in the second will unambiguouslydenote the value in question.

Bernoulli number 3

Explicit definitionStarting again with a slightly more general formula

the choices n = 0 and n = 1 lead to

There is a widespread misinformation that no simple closed formulas for the Bernoulli numbers exist (Gould 1972).The last two equations show that this is not true. Moreover, already in 1893 Louis Saalschütz listed a total of 38explicit formulas for the Bernoulli numbers (Saalschütz 1893), usually giving some reference in the older literature.

Generating functionThe general formula for the generating function is

The choices n = 0 and n = 1 lead to

Algorithmic descriptionAlthough the above recursive formula can be used for computation it is mainly used to establish the connection withthe sum of powers because it is computationally expensive. However, both simple and high-end algorithms forcomputing Bernoulli numbers exist. Pointers to high-end algorithms are given the next section. A simple one is givenin pseudocode below.

Algorithm Akiyama–Tanigawa algorithm for second Bernoulli numbers Bn

Input: Integer n≥0. Output: Second Bernoulli number B

n.

for m from 0 by 1 to n do

A[m] ← 1/(m+1) for j from m by -1 to 1 do

A[j-1] ← j×(A[j-1] - A[j]) return A[0] (which is B

n)

• "←" is a shorthand for "changes to". For instance, "largest ← item" means that the value of largest changes to the value of item.

• "return" terminates the algorithm and outputs the value that follows.

Bernoulli number 4

Efficient computation of Bernoulli numbersIn some applications it is useful to be able to compute the Bernoulli numbers B0 through Bp − 3 modulo p, where p isa prime; for example to test whether Vandiver's conjecture holds for p, or even just to determine whether p is anirregular prime. It is not feasible to carry out such a computation using the above recursive formulae, since at least (aconstant multiple of) p2 arithmetic operations would be required. Fortunately, faster methods have been developed(Buhler et al. 2001) which require only O(p (log p)2) operations (see big-O notation).David Harvey (Harvey 2008) describes an algorithm for computing Bernoulli numbers by computing Bn modulo pfor many small primes p, and then reconstructing Bn via the Chinese Remainder Theorem. Harvey writes that theasymptotic time complexity of this algorithm is O(n2 log(n)2+eps) and claims that this implementation is significantlyfaster than implementations based on other methods. Using this implementation Harvey computed Bn for n = 108.Harvey's implementation is included in Sage since version 3.1. Pavel Holoborodko (Holoborodko 2012) computedBn for n = 2*108 using Harvey's implementation, which is a new record. Prior to that Bernd Kellner (Kellner 2002)computed Bn to full precision for n = 106 in December 2002 and Oleksandr Pavlyk (Pavlyk 2008) for n = 107 with'Mathematica' in April 2008.

Computer Year n Digits*

J. Bernoulli ~1689 10 1

L. Euler 1748 30 8

J.C. Adams 1878 62 36

D.E. Knuth, T.J. Buckholtz 1967 1672 3330

G. Fee, S. Plouffe 1996 10000 27677

G. Fee, S. Plouffe 1996 100000 376755

B.C. Kellner 2002 1000000 4767529

O. Pavlyk 2008 10000000 57675260

D. Harvey 2008 100000000 676752569

P. Holoborodko 2012 200000000

• Digits is to be understood as the exponent of 10 when B(n) is written as a real in normalized scientific notation.

Different viewpoints and conventionsThe Bernoulli numbers can be regarded from four main viewpoints:•• as standalone arithmetical objects,•• as combinatorial objects,•• as values of a sequence of certain polynomials,•• as values of the Riemann zeta function.Each of these viewpoints leads to a set of more or less different conventions.Bernoulli numbers as standalone arithmetical objects.Associated sequence: 1/6, −1/30, 1/42, −1/30, …This is the viewpoint of Jakob Bernoulli. (See the cutout from his Ars Conjectandi, first edition, 1713). TheBernoulli numbers are understood as numbers, recursive in nature, invented to solve a certain arithmetical problem,the summation of powers, which is the paradigmatic application of the Bernoulli numbers. These are also thenumbers appearing in the Taylor series expansion of tan(x) and tanh(x). It is misleading to call this viewpoint'archaic'. For example Jean-Pierre Serre uses it in his highly acclaimed book A Course in Arithmetic which is astandard textbook used at many universities today.

Bernoulli number 5

Bernoulli numbers as combinatorial objects.Associated sequence: 1, +1/2, 1/6, 0, …This view focuses on the connection between Stirling numbers and Bernoulli numbers and arises naturally in thecalculus of finite differences. In its most general and compact form this connection is summarized by the definitionof the Stirling polynomials σn(x), formula (6.52) in Concrete Mathematics by Graham, Knuth and Patashnik.

In consequence Bn = n! σn(1) for n ≥ 0.Bernoulli numbers as values of a sequence of certain polynomials.Assuming the Bernoulli polynomials as already introduced the Bernoulli numbers can be defined in two differentways:• Bn = Bn(0). Associated sequence: 1, −1/2, 1/6, 0, …• Bn = Bn(1). Associated sequence: 1, +1/2, 1/6, 0, …The two definitions differ only in the sign of B1. The choice Bn = Bn(0) is the convention used in the Handbook ofMathematical Functions.

The Bernoulli numbers as given by the Riemann zeta function.

Bernoulli numbers as values of the Riemann zetafunction.Associated sequence: 1, +1/2, 1/6, 0, …Using this convention, the values of the Riemann zetafunction satisfy nζ(1 − n) = −Bn for all integers n≥0.(See the paper of S. C. Woon; the expression nζ(1 − n)for n = 0 is to be understood as limx → 0 xζ(1 − x).)

Applications of the Bernoullinumbers

Asymptotic analysis

Arguably the most important application of the Bernoulli number in mathematics is their use in theEuler–MacLaurin formula. Assuming that ƒ is a sufficiently often differentiable function the Euler–MacLaurinformula can be written as [4]

This formulation assumes the convention B1 = −1/2. Using the convention B1 = 1/2 the formula becomes

Here ƒ(0) = ƒ which is a commonly used notation identifying the zero-th derivative of ƒ with ƒ. Moreover, let ƒ(−1)

denote an antiderivative of ƒ. By the fundamental theorem of calculus,

Thus the last formula can be further simplified to the following succinct form of the Euler–Maclaurin formula

Bernoulli number 6

This form is for example the source for the important Euler–MacLaurin expansion of the zeta function (B1 = 1⁄2)

Here denotes the rising factorial power.[5]

Bernoulli numbers are also frequently used in other kinds of asymptotic expansions. The following example is theclassical Poincaré-type asymptotic expansion of the digamma function (again B1 = 1⁄2).

Taylor series of tan and tanhThe Bernoulli numbers appear in the Taylor series expansion of the tangent and the hyperbolic tangent functions:

Use in topologyThe Kervaire–Milnor formula for the order of the cyclic group of diffeomorphism classes of exotic (4n − 1)-sphereswhich bound parallelizable manifolds involves Bernoulli numbers. Let ESn be the number of such exotic spheres forn ≥ 2, then

The Hirzebruch signature theorem for the L genus of a smooth oriented closed manifold of dimension 4n alsoinvolves Bernoulli numbers.

Combinatorial definitionsThe connection of the Bernoulli number to various kinds of combinatorial numbers is based on the classical theoryof finite differences and on the combinatorial interpretation of the Bernoulli numbers as an instance of a fundamentalcombinatorial principle, the inclusion-exclusion principle.

Connection with Worpitzky numbersThe definition to proceed with was developed by Julius Worpitzky in 1883. Besides elementary arithmetic only thefactorial function n! and the power function km is employed. The signless Worpitzky numbers are defined as

They can also be expressed through the Stirling numbers of the second kind

Bernoulli number 7

A Bernoulli number is then introduced as an inclusion-exclusion sum of Worpitzky numbers weighted by thesequence 1, 1/2, 1/3, …

This representation has B1 = 1/2.

Worpitzky's representation of the Bernoulli number

B0 = 1/1

B1 = 1/1 − 1/2

B2 = 1/1 − 3/2 + 2/3

B3 = 1/1 − 7/2 + 12/3 − 6/4

B4 = 1/1 − 15/2 + 50/3 − 60/4 + 24/5

B5 = 1/1 − 31/2 + 180/3 − 390/4 + 360/5 − 120/6

B6 = 1/1 − 63/2 + 602/3 − 2100/4 + 3360/5 − 2520/6 + 720/7

A second formula representing the Bernoulli numbers by the Worpitzky numbers is for n ≥ 1

Connection with Stirling numbers of the second kind

If denotes Stirling numbers of the second kind[6] then one has:

where denotes the falling factorial.

If one defines the Bernoulli polynomials as[7]:

where for are the Bernoulli numbers.Then after the following property of binomial coefficient:

one has,

One also has following for Bernoulli polynomials,[7]

The coefficient of j in is Comparing the coefficient of j in the two expressions of Bernoulli polynomials, one has:

Bernoulli number 8

(resulting in B1=1/2) which is an explicit formula for Bernoulli numbers and can be used to prove Von-StaudtClausen theorem.[8][9][10]

Connection with Stirling numbers of the first kind

The two main formulas relating the unsigned Stirling numbers of the first kind to the Bernoulli numbers (with

B1 = 1/2) are

and the inversion of this sum (for n ≥ 0, m ≥ 0)

Here the number An,m are the rational Akiyama–Tanigawa numbers, the first few of which are displayed in thefollowing table.

Akiyama–Tanigawa number

n \ m 0 1 2 3 4

0 1 1/2 1/3 1/4 1/5

1 1/2 1/3 1/4 1/5 ...

2 1/6 1/6 3/20 ... ...

3 0 1/30 ... ... ...

4 −1/30 ... ... ... ...

The Akiyama–Tanigawa numbers satisfy a simple recurrence relation which can be exploited to iteratively computethe Bernoulli numbers. This leads to the algorithm shown in the section 'algorithmic description' above.

Connection with Eulerian numbers

There are formulas connecting Eulerian numbers to Bernoulli numbers:

Both formulas are valid for n ≥ 0 if B1 is set to ½. If B1 is set to −½ they are valid only for n ≥ 1 and n ≥ 2respectively.

Bernoulli number 9

Connection with Balmer seriesA link between Bernoulli numbers and Balmer series could be seen in sequence A191567 in OEIS.

Representation of the second Bernoulli numbersSee A191302 in OEIS. The number are not reduced. Then the columns are easy to find, the denominators beingA190339.

Representation of the second Bernoulli numbers

B0 = 1 = 2/2

B1 = 1/2

B2 = 1/2 − 2/6

B3 = 1/2 − 3/6

B4 = 1/2 − 4/6 +  2/15

B5 = 1/2 − 5/6 +  5/15

B6 = 1/2 − 6/6 +  9/15 −   8/105

B7 = 1/2 − 7/6 + 14/15 − 28/105

A binary tree representationThe Stirling polynomials σn(x) are related to the Bernoulli numbers by Bn = n!σn(1). S. C. Woon (Woon 1997)described an algorithm to compute σn(1) as a binary tree.

Woon's tree for σn(1)

Woon's recursive algorithm (for n ≥ 1) starts by assigning to the root node N = [1,2]. Given a node N = [a1,a2,..., ak]of the tree, the left child of the node is L(N) = [−a1,a2 + 1, a3, ..., ak] and the right child R(N) = [a1,2, a2, ..., ak]. Anode N = [a1,a2,..., ak] is written as ±[a2,..., ak] in the initial part of the tree represented above with ± denoting thesign of a1.

Given a node N the factorial of N is defined as

Restricted to the nodes N of a fixed tree-level n the sum of 1/N! is σn(1), thus

Bernoulli number 10

For example B1 = 1!(1/2!), B2 = 2!(−1/3! + 1/(2!2!)), B3 = 3!(1/4! − 1/(2!3!) − 1/(3!2!) + 1/(2!2!2!)).

Asymptotic approximationLeonhard Euler expressed the Bernoulli numbers in terms of the Riemann zeta function as

It then follows from the Stirling formula that, as n goes to infinity,

Including more terms from the zeta series yields a better approximation, as does factoring in the asymptotic series inStirling's approximation.

Integral representation and continuationThe integral

has as special values b(2n) = B2n for n > 0. The integral might be considered as a continuation of the Bernoullinumbers to the complex plane and this was indeed suggested by Peter Luschny in 2004.For example b(3) = (3/2)ζ(3)Π−3Ι and b(5) = −(15/2) ζ(5) Π −5Ι. Here ζ(n) denotes the Riemann zeta function and Ιthe imaginary unit. It is remarkable that already Leonhard Euler (Opera Omnia, Ser. 1, Vol. 10, p. 351) consideredthese numbers and calculated

Euler's values are unsigned and real, but obviously his aim was to find a meaningful way to define the Bernoullinumbers at the odd integers n > 1.

The relation to the Euler numbers and πThe Euler numbers are a sequence of integers intimately connected with the Bernoulli numbers. Comparing theasymptotic expansions of the Bernoulli and the Euler numbers shows that the Euler numbers E2n are in magnitudeapproximately (2/π)(42n − 22n) times larger than the Bernoulli numbers B2n. In consequence:

This asymptotic equation reveals that π lies in the common root of both the Bernoulli and the Euler numbers. In factπ could be computed from these rational approximations.Bernoulli numbers can be expressed through the Euler numbers and vice versa. Since for n odd Bn = En = 0 (with theexception B1), it suffices to consider the case when n is even.

Bernoulli number 11

These conversion formulas express an inverse relation between the Bernoulli and the Euler numbers. But moreimportant, there is a deep arithmetic root common to both kinds of numbers, which can be expressed through a morefundamental sequence of numbers, also closely tied to π. These numbers are defined for n > 1 as

and S1 = 1 by convention (Elkies 2003). The magic of these numbers lies in the fact that they turn out to be rationalnumbers. This was first proved by Leonhard Euler in a landmark paper (Euler 1735) ‘De summis serierumreciprocarum’ (On the sums of series of reciprocals) and has fascinated mathematicians ever since. The first few ofthese numbers are

The Bernoulli numbers and Euler numbers are best understood as special views of these numbers, selected from thesequence Sn and scaled for use in special applications.

The expression [n even] has the value 1 if n is even and 0 otherwise (Iverson bracket).These identities show that the quotient of Bernoulli and Euler numbers at the beginning of this section is just thespecial case of Rn = 2Sn / Sn+1 when n is even. The Rn are rational approximations to π and two successive termsalways enclose the true value of π. Beginning with n = 1 the sequence starts

These rational numbers also appear in the last paragraph of Euler's paper cited above.

An algorithmic view: the Seidel triangleThe sequence Sn has another unexpected yet important property: The denominators of Sn divide the factorial (n − 1)!.In other words: the numbers Tn = Sn(n − 1)! are integers.

Thus the above representations of the Bernoulli and Euler numbers can be rewritten in terms of this sequence as

These identities make it easy to compute the Bernoulli and Euler numbers: the Euler numbers En are givenimmediately by T2n + 1 and the Bernoulli numbers B2n are obtained from T2n by some easy shifting, avoiding rationalarithmetic.What remains is to find a convenient way to compute the numbers Tn. However, already in 1877 Philipp Ludwig vonSeidel (Seidel 1877) published an ingenious algorithm which makes it extremely simple to calculate Tn.

Bernoulli number 12

Seidel's algorithm for Tn

[begin] Start by putting 1 in row 0 and let k denote the number of the row currently being filled. If k is odd, then putthe number on the left end of the row k − 1 in the first position of the row k, and fill the row from the left to the right,with every entry being the sum of the number to the left and the number to the upper. At the end of the row duplicatethe last number. If k is even, proceed similar in the other direction. [end]Seidel's algorithm is in fact much more general (see the exposition of Dominique Dumont (Dumont 1981)) and wasrediscovered several times thereafter.Similar to Seidel's approach D. E. Knuth and T. J. Buckholtz (Knuth & Buckholtz 1967) gave a recurrence equationfor the numbers T2n and recommended this method for computing B2n and E2n ‘on electronic computers using onlysimple operations on integers’.V. I. Arnold rediscovered Seidel's algorithm in (Arnold 1991) and later Millar, Sloane and Young popularizedSeidel's algorithm under the name boustrophedon transform.

A combinatorial view: alternating permutationsAround 1880, three years after the publication of Seidel's algorithm, Désiré André proved a now classic result ofcombinatorial analysis (André 1879) & (André 1881). Looking at the first terms of the Taylor expansion of thetrigonometric functions tan x and sec x André made a startling discovery.

The coefficients are the Euler numbers of odd and even index, respectively. In consequence the ordinary expansionof tan x + sec x has as coefficients the rational numbers Sn.

André then succeeded by means of a recurrence argument to show that the alternating permutations of odd size areenumerated by the Euler numbers of odd index (also called tangent numbers) and the alternating permutations ofeven size by the Euler numbers of even index (also called secant numbers).

Bernoulli number 13

Related sequencesThe arithmetic mean of the first and the second Bernoulli numbers are the associate Bernoulli numbers: B0 = 1, B1 =0, B2 = 1/6, B3 = 0, B4 = -1/30, A176327 / A027642 in OEIS. Via the second row of its inverse Akiyama-Tanigawatransform (sequence A177427 in OEIS), they lead to Balmer series A061037 / A061038.

A companion to the second Bernoulli numbersSee A190339. These numbers are the eigensequence of the first kind. A191754 / A192366 = 0, 1/2, 1/2, 1/3, 1/6,1/15, 1/30, 1/35, 1/70, -1/105, -1/210, 41/1155, 41/2310, -589/5005, -589/10010 ...

Arithmetical properties of the Bernoulli numbersThe Bernoulli numbers can be expressed in terms of the Riemann zeta function as Bn = − nζ(1 − n) for integers n ≥ 0provided for n = 0 and n = 1 the expression − nζ(1 − n) is understood as the limiting value and the convention B1 =1/2 is used. This intimately relates them to the values of the zeta function at negative integers. As such, they could beexpected to have and do have deep arithmetical properties. For example, the Agoh–Giuga conjecture postulates thatp is a prime number if and only if pBp−1 is congruent to −1 modulo p. Divisibility properties of the Bernoullinumbers are related to the ideal class groups of cyclotomic fields by a theorem of Kummer and its strengthening inthe Herbrand-Ribet theorem, and to class numbers of real quadratic fields by Ankeny–Artin–Chowla.

The Kummer theoremsThe Bernoulli numbers are related to Fermat's last theorem (FLT) by Kummer's theorem (Kummer 1850), whichsays:If the odd prime p does not divide any of the numerators of the Bernoulli numbers B2, B4, ..., Bp−3 thenxp + yp + zp = 0 has no solutions in non-zero integers.Prime numbers with this property are called regular primes. Another classical result of Kummer (Kummer 1851) arethe following congruences.Let p be an odd prime and b an even number such that p − 1 does not divide b. Then for any non-negative integer k

A generalization of these congruences goes by the name of p-adic continuity.

p-adic continuityIf b, m and n are positive integers such that m and n are not divisible by p − 1 and , then

Since Bn = —n ζ(1 — n), this can also be written

where u = 1 − m and v = 1 − n, so that u and v are nonpositive and not congruent to 1 modulo p − 1. This tells us thatthe Riemann zeta function, with 1 − p−s taken out of the Euler product formula, is continuous in the p-adic numberson odd negative integers congruent modulo p − 1 to a particular , and so can be extended to acontinuous function ζp(s) for all p-adic integers , the p-adic zeta function.

Bernoulli number 14

Ramanujan's congruencesThe following relations, due to Ramanujan, provide a method for calculating Bernoulli numbers that is moreefficient than the one given by their original recursive definition:

Von Staudt–Clausen theoremThe von Staudt–Clausen theorem was given by Karl Georg Christian von Staudt (von Staudt 1840) and ThomasClausen (Clausen 1840) independently in 1840. The theorem states that for every n > 0,

is an integer. The sum extends over all primes p for which p − 1 divides 2n.A consequence of this is that the denominator of B2n is given by the product of all primes p for which p − 1 divides2n. In particular, these denominators are square-free and divisible by 6.

Why do the odd Bernoulli numbers vanish?The sum

can be evaluated for negative values of the index n. Doing so will show that it is an odd function for even values ofk, which implies that the sum has only terms of odd index. This and the formula for the Bernoulli sum imply thatB2k+1−m is 0 for m odd and greater than 1; and that the term for B1 is cancelled by the subtraction. Thevon Staudt Clausen theorem combined with Worpitzky's representation also gives a combinatorial answer to thisquestion (valid for n > 1).From the von Staudt Clausen theorem it is known that for odd n > 1 the number 2Bn is an integer. This seems trivialif one knows beforehand that in this case Bn = 0. However, by applying Worpitzky's representation one gets

as a sum of integers, which is not trivial. Here a combinatorial fact comes to surface which explains the vanishing ofthe Bernoulli numbers at odd index. Let Sn,m be the number of surjective maps from {1, 2, ..., n} to {1, 2, ..., m}, then

. The last equation can only hold if

This equation can be proved by induction. The first two examples of this equation aren = 4:  2 + 8 = 7 + 3,n = 6:  2 + 120 + 144 = 31 + 195 + 40.

Thus the Bernoulli numbers vanish at odd index because some non-obvious combinatorial identities are embodied inthe Bernoulli numbers.

Bernoulli number 15

A restatement of the Riemann hypothesisThe connection between the Bernoulli numbers and the Riemann zeta function is strong enough to provide analternate formulation of the Riemann hypothesis (RH) which uses only the Bernoulli number. In fact Marcel Riesz(Riesz 1916) proved that the RH is equivalent to the following assertion:For every ε > 1/4 there exists a constant Cε > 0 (depending on ε) such that |R(x)| < Cε x

ε as x → ∞.Here R(x) is the Riesz function

denotes the rising factorial power in the notation of D. E. Knuth. The number βn = Bn/n occur frequently in thestudy of the zeta function and are significant because βn is a p-integer for primes p where p − 1 does not divide n.The βn are called divided Bernoulli number.

History

Early historyThe Bernoulli numbers are rooted in the early history of the computation of sums of integer powers, which havebeen of interest to mathematicians since antiquity.

A page from Seki Kōwa's Katsuyo Sampo (1712),tabulating binomial coefficients and Bernoulli

numbers

Methods to calculate the sum of the first n positive integers, the sum ofthe squares and of the cubes of the first n positive integers wereknown, but there were no real 'formulas', only descriptions givenentirely in words. Among the great mathematicians of antiquity whichconsidered this problem were: Pythagoras (c. 572–497 BCE, Greece),Archimedes (287–212 BCE, Italy), Aryabhata (b. 476, India), AbuBakr al-Karaji (d. 1019, Persia) and Abu Ali al-Hasan ibn al-Hasan ibnal-Haytham (965–1039, Iraq).

During the late sixteenth and early seventeenth centuriesmathematicians made significant progress. In the West Thomas Harriot(1560–1621) of England, Johann Faulhaber (1580–1635) of Germany,Pierre de Fermat (1601–1665) and fellow French mathematician BlaisePascal (1623–1662) all played important roles.

Thomas Harriot seems to have been the first to derive and writeformulas for sums of powers using symbolic notation, but even hecalculated only up to the sum of the fourth powers. Johann Faulhabergave formulas for sums of powers up to the 17th power in his 1631Academia Algebrae, far higher than anyone before him, but he did notgive a general formula.

The Swiss mathematician Jakob Bernoulli (1654–1705) was the first to realize the existence of a single sequence ofconstants B0, B1, B2, ... which provide a uniform formula for all sums of powers (Knuth 1993).The joy Bernoulli experienced when he hit upon the pattern needed to compute quickly and easily the coefficients ofhis formula for the sum of the c-th powers for any positive integer c can be seen from his comment. He wrote:“With the help of this table, it took me less than half of a quarter of an hour to find that the tenth powers of the first1000 numbers being added together will yield the sum91,409,924,241,424,243,424,241,924,242,500.”

Bernoulli number 16

Bernoulli's result was published posthumously in Ars Conjectandi in 1713. Seki Kōwa independently discovered theBernoulli numbers and his result was published a year earlier, also posthumously, in 1712.[1] However, Seki did notpresent his method as a formula based on a sequence of constants.Bernoulli's formula for sums of powers is the most useful and generalizable formulation to date. The coefficients inBernoulli's formula are now called Bernoulli numbers, following a suggestion of Abraham de Moivre.Bernoulli's formula is sometimes called Faulhaber's formula after Johann Faulhaber who found remarkable ways tocalculate sum of powers but never stated Bernoulli's formula. To call Bernoulli's formula Faulhaber's formula doesinjustice to Bernoulli and simultaneously hides the genius of Faulhaber as Faulhaber's formula is in fact moreefficient than Bernoulli's formula. According to Knuth (Knuth 1993) a rigorous proof of Faulhaber’s formula wasfirst published by Carl Jacobi in 1834 (Jacobi 1834). Donald E. Knuth's in-depth study of Faulhaber's formulaconcludes:“Faulhaber never discovered the Bernoulli numbers; i.e., he never realized that a single sequence of constants B0,B1, B2, ... would provide a uniform

for all sums of powers. He never mentioned, for example, the fact that almost half of the coefficients turned out to bezero after he had converted his formulas for from polynomials in N to polynomials in n.” (Knuth 1993, p. 14)

Reconstruction of "Summae Potestatum"

Jakob Bernoulli's Summae Potestatum, 1713

The Bernoulli numbers were introduced by Jakob Bernoulli in the bookArs Conjectandi published posthumously in 1713. The main formulacan be seen in the second half of the corresponding facsimile. Theconstant coefficients denoted A, B, C and D by Bernoulli are mappedto the notation which is now prevalent as A = B2, B = B4, C = B6,D = B8. In the expression c·c−1·c−2·c−3 the small dots are used asgrouping symbols, not as signs for multiplication. Using today'sterminology these expressions are falling factorial powers . Thefactorial notation k! as a shortcut for 1 × 2 × ... × k was not introduceduntil 100 years later. The integral symbol on the left hand side goesback to Gottfried Wilhelm Leibniz in 1675 who used it as a long letterS for "summa" (sum). (The Mathematics Genealogy Project [11] showsLeibniz as the doctoral adviser of Jakob Bernoulli. See also the EarliestUses of Symbols of Calculus.[12]) The letter n on the left hand side isnot an index of summation but gives the upper limit of the range ofsummation which is to be understood as 1, 2, …, n. Putting thingstogether, for positive c, today a mathematician is likely to write Bernoulli's formula as:

In fact this formula imperatively suggests to set B1 = ½ when switching from the so-called 'archaic' enumerationwhich uses only the even indices 2, 4, … to the modern form (more on different conventions in the next paragraph).Most striking in this context is the fact that the falling factorial has for k = 0 the value .[13] Thus Bernoulli'sformula can and has to be written:

Bernoulli number 17

If B1 stands for the value Bernoulli himself has given to the coefficient at that position.

Generalized Bernoulli numbersThe generalized Bernoulli numbers are certain algebraic numbers, defined similarly to the Bernoulli numbers, thatare related to special values of Dirichlet L-functions in the same way that Bernoulli numbers are related to specialvalues of the Riemann zeta function.Let χ be a primitive Dirichlet character modulo f. The generalized Bernoulli numbers attached to χ are defined by

Let ε ∈ {0, 1} be defined by χ(−1) = (−1)ε. Then,Bk,χ ≠ 0 if, and only if, k ≡ ε (mod 2).

Generalizing the relation between Bernoulli numbers and values of the Riemann zeta function at non-positiveintegers, one has the for all integers k ≥ 1

where L(s, χ) is the Dirichlet L-function of χ.[14]

Appendix

Assorted identities• Umbral calculus gives a compact form of Bernoulli's formula by using an abstract symbol B:

where the symbol that appears during binomial expansion of the parenthesized term is to be replaced by theBernoulli number (and ). More suggestively and mnemonically, this may be written as a definiteintegral:

Many other Bernoulli identities can be written compactly with this symbol, e.g.

• Let n be non-negative and even

• The nth cumulant of the uniform probability distribution on the interval [−1, 0] is Bn/n.• Let n¡ = 1/n! and n ≥ 1. Then Bn is the following determinant:

Bernoulli number 18

Bn =1

───

n¡

2¡ 1¡ 0 0 0 ... 0

3¡ 2¡ 1¡ 0 0 ... 0

4¡ 3¡ 2¡ 1¡ 0 ... 0

... ... ... ... ... ... ...

(n − 1)¡ ... ... 3¡ 2¡ 1¡ 0

n¡ (n − 1)¡ ... ... 3¡ 2¡ 1¡

(n + 1)¡ n¡ (n − 1)¡ ... ... 3¡ 2¡

Thus the determinant is σn(1), the Stirling polynomial at x = 1.• For even-numbered Bernoulli numbers, B2p is given by the p X p determinant:[15]

B2p =

(−1)p+1

───

(22p − 2)(2p)¡

3¡ 1¡ 0 0 0 ... 0

5¡ 3¡ 1¡ 0 0 ... 0

7¡ 5¡ 3¡ 1¡ 0 ... 0

... ... ... ... ... ... ...

(2p − 3)¡ ... ... 5¡ 3¡ 1¡ 0

(2p − 1)¡ (2p − 3)¡ ... ... 5¡ 3¡ 1¡

(2p + 1)¡ (2p − 1)¡ (2p − 3)¡ ... ... 5¡ 3¡

• Let n ≥ 1.

• Let n ≥ 1. Then (von Ettingshausen 1827)

• Let n ≥ 0. Then (Leopold Kronecker 1883)

• Let n ≥ 1 and m ≥ 1. Then (Carlitz 1968)

• Let n ≥ 4 and

the harmonic number. Then

• Let n ≥ 4. Yuri Matiyasevich found (1997)

Bernoulli number 19

• Faber-Pandharipande-Zagier-Gessel identity: for n ≥ 1,

Choosing x = 0 or x = 1 results in the Bernoulli number identity in one or another convention.• The next formula is true for n ≥ 0 if B1 = B1(1) = ½, but only for n ≥ 1 if B1 = B1(0) = −½.

• Let n ≥ 0 and [b] = 1 if b is true, 0 otherwise.

and

Values of the first Bernoulli numbersBn = 0 for all odd n other than 1. For even n, Bn is negative if n is divisible by 4 and positive otherwise. The first fewnon-zero Bernoulli numbers are:

n Numerator Denominator Decimal approximation

0 1 1 +1.00000000000

1 −1 2 −0.50000000000

2 1 6 +0.16666666667

4 −1 30 −0.03333333333

6 1 42 +0.02380952381

8 −1 30 −0.03333333333

10 5 66 +0.07575757576

12 −691 2730 −0.25311355311

14 7 6 +1.16666666667

16 −3617 510 −7.09215686275

18 43867 798 +54.9711779448

OEIS A027641 A027642

From 6, the denominators are multiples of the sequence of period 2 : 6,30 (sequence A165734 in OEIS). From 2, thedenominators are of the form 4*k + 2.

Bernoulli number 20

Notes[1][1] Selin, H. (1997), p. 891[2][2] Smith, D. E. (1914), p. 108[3] Note G in the Menabrea reference[4] Concrete Mathematics, (9.67).[5] Concrete Mathematics, (2.44) and (2.52)[6][6] L. Comtet, Advanced combinatorics. The art of finite and infinite expansions, Revised and Enlarged Edition, D. Reidel Publ. Co.,

Dordrecht-Boston, 1974.[7][7] H. Rademacher, Analytic Number Theory, Springer-Verlag, New York, 1973.[8] H. W. Gould (1972). "Explicit formulas for Bernoulli numbers". Amer. Math. Monthly 79: 44–51.[9] T. M. Apostol. Introduction to Analytic Number Theory. Springer-Verlag. p. 197.[10] G. Boole (1880). A treatise of the calculus of finite differences (3rd ed ed.). London.[11] Mathematics Genealogy Project (http:/ / genealogy. math. ndsu. nodak. edu/ )[12] Earliest Uses of Symbols of Calculus (http:/ / jeff560. tripod. com/ calculus. html)[13] Graham, R.; Knuth, D. E.; Patashnik, O. (1989), Concrete Mathematics (2nd ed.), Addison-Wesley, Section 2.51, ISBN 0-201-55802-5[14][14] Neukirch 1999, §VII.2[15][15] Jerome Malenfant (2011). "Finite, closed-form expressions for the partition function and for Euler, Bernoulli, and Stirling numbers".

arXiv:1103.1585 [math.NT].

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Bernoulli number 21

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Bernoulli number 22

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External links• Hazewinkel, Michiel, ed. (2001), "Bernoulli numbers" (http:/ / www. encyclopediaofmath. org/ index.

php?title=p/ b015640), Encyclopedia of Mathematics, Springer, ISBN 978-1-55608-010-4• The first 498 Bernoulli Numbers (http:/ / www. gutenberg. org/ etext/ 2586) from Project Gutenberg• A multimodular algorithm for computing Bernoulli numbers (http:/ / web. maths. unsw. edu. au/ ~davidharvey/

papers/ bernmm/ )• The Bernoulli Number Page (http:/ / www. bernoulli. org)• Bernoulli number programs (http:/ / en. literateprograms. org/ Category:Bernoulli_numbers) at LiteratePrograms

(http:/ / en. literateprograms. org)• Weisstein, Eric W., " Bernoulli Number (http:/ / mathworld. wolfram. com/ BernoulliNumber. html)" from

MathWorld.• The Computation of Irregular Primes (P. Luschny) (http:/ / www. luschny. de/ math/ primes/ irregular. html)• The Computation And Asymptotics Of Bernoulli Numbers (P. Luschny) (http:/ / oeis. org/ wiki/

User:Peter_Luschny/ ComputationAndAsymptoticsOfBernoulliNumbers)• Bernoullinumbers in context of Pascal-(Binomial)matrix (http:/ / go. helms-net. de/ math/ pascal/ bernoulli_en.

pdf) german version (http:/ / go. helms-net. de/ math/ pascal/ bernoulli. pdf)• summing of like powers in context with Pascal-/Bernoulli-matrix (http:/ / go. helms-net. de/ math/ binomial/

04_3_SummingOfLikePowers. pdf)• Some special properties, sums of Bernoulli-and related numbers (http:/ / go. helms-net. de/ math/ binomial/

02_2_GeneralizedBernoulliRecursion. pdf)• Bernoulli Numbers Calculator (http:/ / www. numberempire. com/ bernoullinumbers. php)

Article Sources and Contributors 23

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