from calculus to number theory · from calculus to number theory author: a. raghuram created date:...

From Calculus to Number Theory

A. Raghuram

IISER Pune

November 4, 2016

Integers

Integers

,,Die ganzen Zahlen hat der liebe Gott gemacht, alles andere istMenschenwerk.”

Integers

,,Die ganzen Zahlen hat der liebe Gott gemacht, alles andere istMenschenwerk.” –Leopold Kronecker (1823-1891)

Integers


(“God made the integers, all else is the work of man.”)

Integers


(“God made the integers, all else is the work of man.”)

Integers = Z := {. . . ,−3,−2,−1, 0, 1, 2, 3, . . . }

A Fourteenth Century theorem


The Harmonic series

1 +1

2+

1

3+

1

4+ · · ·+ 1

n+ · · ·


The Harmonic series

1 +1

2+

1

3+

1

4+ · · ·+ 1

n+ · · ·

diverges to infinity.


The Harmonic series

1 +1

2+

1

3+

1

4+ · · ·+ 1

n+ · · ·


This means given any number M, however large, we can addsufficiently many terms in the above series to make the sum largerthan M.


The Harmonic series

1 +1

2+

1

3+

1

4+ · · ·+ 1

n+ · · ·


This means given any number M, however large, we can addsufficiently many terms in the above series to make the sum largerthan M.

This was first proved by Nicole Oresme (1323-1382), a brilliantFrench philosopher of his times.

The Basel Problem

The Basel Problem

The series

1 +1

22+

1

32+

1

42+ · · ·+ 1

n2+ · · ·

The Basel Problem

The series

1 +1

22+

1

32+

1

42+ · · ·+ 1

n2+ · · ·

converges! (Compare with∫∞1

1x2

dx .)

The Basel Problem

The series

1 +1

22+

1

32+

1

42+ · · ·+ 1

n2+ · · ·


1x2

dx .)

The question is what does it converge to? Meaning, what is thesum of this series?

The Basel Problem

The series

1 +1

22+

1

32+

1

42+ · · ·+ 1

n2+ · · ·


1x2

dx .)

The question is what does it converge to? Meaning, what is thesum of this series?

This went by the name “The Basel Problem”, posed first by PietroMengoli in 1644, and was unsuccessfully–but famously–attacked bythe Bernoulli family.

A kid in his twenties solved the Basel problem in 1735


Leonard Euler (1707-1783)



1 +1

22+

1

32+

1

42+ · · ·+ 1

n2+ · · · =

π2

6



1 +1

22+

1

32+

1

42+ · · ·+ 1

n2+ · · · =

π2

6

Suggested reading: A journey through genius: great theorems of Mathematics, by William Dunham, published byPenguin.

Another Fourteenth Century theorem


The alternating series

1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · ·



1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · ·

converges.



1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · ·

converges. In fact, we have:



1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · ·


1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · · =

π

4



1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · ·


1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · · =

π

4

This was first proved by Madhava (c. 1350 - c. 1425) from theKerala School of Mathematics.



1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · ·


1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · · =

π

4


Later it was independently arrived at by Leibniz (1646-1716) andGregory (1638-1675).



1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · ·


1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · · =

π

4


Later it was independently arrived at by Leibniz (1646-1716) andGregory (1638-1675).

Proof: Put x = 1 in the expansion

tan−1(x) = x − x3

3+

x5

5− x7

7+ · · ·

Prototypical examples


Euler’s formula

1 +1

22+

1

32+

1

42+ · · ·+ 1

n2+ · · · =

π2

6


Euler’s formula

1 +1

22+

1

32+

1

42+ · · ·+ 1

n2+ · · · =

π2

6

and Madhava-Gregory-Leibniz series

1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · · =

π

4


Euler’s formula

1 +1

22+

1

32+

1

42+ · · ·+ 1

n2+ · · · =

π2

6

and Madhava-Gregory-Leibniz series

1− 1

3+

1

5− 1

7+ · · ·+ (−1)n

2n + 1+ · · · =

π

4

are prototypical examples of an exciting area of research in modernnumber theory that goes by the appellation:

Special Values of L-functions.

The Riemann zeta function


ζ(s) := 1 +1

2s+

1

3s+

1

4s+ · · ·+ 1

ns+ · · ·


ζ(s) := 1 +1

2s+

1

3s+

1

4s+ · · ·+ 1

ns+ · · ·

Riemann studied this function of a complex variable s in alandmark paper in 1859.


ζ(s) := 1 +1

2s+

1

3s+

1

4s+ · · ·+ 1

ns+ · · ·

Riemann studied this function of a complex variable s in alandmark paper in 1859. He introduced this function to study thedistribution of primes.


ζ(s) := 1 +1

2s+

1

3s+

1

4s+ · · ·+ 1

ns+ · · ·


Bernhard Riemann (1826-1846)


ζ(s) := 1 +1

2s+

1

3s+

1

4s+ · · ·+ 1

ns+ · · ·



Euler’s formula may be stated as ζ(2) = π2

6 .


ζ(s) := 1 +1

2s+

1

3s+

1

4s+ · · ·+ 1

ns+ · · ·




6 . It was proved byApery in 1979 that ζ(3) is irrational.


ζ(s) := 1 +1

2s+

1

3s+

1

4s+ · · ·+ 1

ns+ · · ·




6 . It was proved byApery in 1979 that ζ(3) is irrational. More generally, ζ(2m) is verywell-understood but ζ(2m + 1) is a total mystery.

Dirichlet L-functions


Take an integer N, and consider the group (Z/NZ)× of integersrelatively prime to N.


Take an integer N, and consider the group (Z/NZ)× of integersrelatively prime to N. Take a homomorphism χ : (Z/NZ)× → C

×.



×.Dirichlet studied the function

L(s, χ) := 1 +χ(2)

2s+

χ(3)

3s+

χ(4)

4s+ · · ·+ χ(n)

ns+ · · ·




L(s, χ) := 1 +χ(2)

2s+

χ(3)

3s+

χ(4)

4s+ · · ·+ χ(n)

ns+ · · ·

to prove that there are infinitely many primes in arithmeticprogressions of the form a, a+ N, a+ 2N, a+ 3N, . . . .




L(s, χ) := 1 +χ(2)

2s+

χ(3)

3s+

χ(4)

4s+ · · ·+ χ(n)

ns+ · · ·


Lejuene Dirichlet (1801-1859)




L(s, χ) := 1 +χ(2)

2s+

χ(3)

3s+

χ(4)

4s+ · · ·+ χ(n)

ns+ · · ·


Lejuene Dirichlet (1801-1859)

Madhava’s formula may be stated as L(1, χ) = π4 if χ is the unique

nontrivial character modulo N = 4.

General L-functions

General L-functions

There are various mathematical objects M that determine, and aredetermined, by a sequence an = an(M).

General L-functions


A general principle in number theory: study M by studying:

General L-functions



L(s,M) := 1 +a2(M)

2s+

a3(M)

3s+

a4(M)

4s+ · · ·+ an(M)

ns+ · · ·

General L-functions



L(s,M) := 1 +a2(M)

2s+

a3(M)

3s+

a4(M)

4s+ · · ·+ an(M)

ns+ · · ·

The basic problems are:

1 Study L(s,M) as a function of a complex variable.

2 Study the values of L(s,M). These provide structuralinformation about M.

General L-functions



L(s,M) := 1 +a2(M)

2s+

a3(M)

3s+

a4(M)

4s+ · · ·+ an(M)

ns+ · · ·




Examples:

General L-functions



L(s,M) := 1 +a2(M)

2s+

a3(M)

3s+

a4(M)

4s+ · · ·+ an(M)

ns+ · · ·




Examples: Dirichlet’s theorem on infinitude of primes in AP boilsdown to proving L(1, χ) �= 0.

General L-functions



L(s,M) := 1 +a2(M)

2s+

a3(M)

3s+

a4(M)

4s+ · · ·+ an(M)

ns+ · · ·




Examples: Dirichlet’s theorem on infinitude of primes in AP boilsdown to proving L(1, χ) �= 0. The value at s = 1 of the Dedekindzeta function of a number field has information about importantinvariants about that number field–the class number formula!

The Langlands Program


In the 1960’s Robert Langlands formulated a web conjecturesconnecting different areas of mathematics: Number Theory,Representation Theory, Harmonic Analysis, Algebraic Geometry, ...



Robert Langlands (Born. 1936)



Robert Langlands (Born. 1936)The basic mathematical idea that connects these different subjectsis that of an L-function.



Robert Langlands (Born. 1936)The basic mathematical idea that connects these different subjectsis that of an L-function. (Think of Andrew Wiles’s celebratedtheorem that every elliptic curve is modular.)

My research-I

My research-I

Theorem (With Baskar Balasubramanyam; to appear in AmericanJournal of Math.)

My research-I


Let σ be a cuspidal automorphic representation of GLn over anumber field.

My research-I



1 We identify periods p(σ) and q(σ) such that

My research-I




Lalg(1,Ad , σ) := L(1,Ad , σ)/p(σ)q(σ)

is algebraic.

My research-I




Lalg(1,Ad , σ) := L(1,Ad , σ)/p(σ)q(σ)

is algebraic.

2 If a prime p divides Lalg(1,Ad , σ) then p is a congruenceprime for σ.

My research-II

My research-II

Theorem (With Chandrasheel Bhagwat)

My research-II


Let σ be a cuspidal automorphic representation of SO(n, n) over atotally real number field.

My research-II


Let σ be a cuspidal automorphic representation of SO(n, n) over atotally real number field. Suppose m and m + 1 are critical pointsfor the standard degree-2n Langlands L-function L(s, σ) attachedto σ.

My research-II


Let σ be a cuspidal automorphic representation of SO(n, n) over atotally real number field. Suppose m and m + 1 are critical pointsfor the standard degree-2n Langlands L-function L(s, σ) attachedto σ. Then

L(m, σ) ≈ L(m + 1, σ),

i.e., two successive L-values differ only by an algebraic number.

Thank You!

from calculus to number theory · from calculus to number theory author: a. raghuram created date:...

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