The Prime Number Theorem - Massachusetts Institute of Technology

The Prime Number Theorem

A PRIMES Exposition

Ishita Goluguri, Toyesh Jayaswal, Andrew Lee Mentor: Chengyang Shao

TABLE OF CONTENTS

Introduction Tools from Complex Analysis Entire Functions Hadamard Factorization Theorem Riemann Zeta Function 6 Chebyshev Functions Perron Formula 8 Prime Number Theorem

? Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao

Introduction

? Euclid ( BC): There are infinitely many primes ? Legendre ( 8 8): for primes less than , , :

(x) x log x

? Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao

Progress on the Distribution of Prime Numbers

? Euler: The product formula

1

1

(s) :=

ns =

1 - p-s

n=1

p

so (heuristically)

1 = log

1 - p-1

p

? Chebyshev ( 8 8- 8 ): if the ratio of (x) and x/ log x has a limit, it must be 1

? Riemann ( 8 ): On the Number of Primes Less Than a Given Magnitude, related (x) to the zeros of (s) using complex analysis

? Hadamard, de la Vall?e Poussin ( 8 6): Proved independently the prime number theorem by showing (s) has no zeros of the form 1 + it, hence the celebrated prime number theorem

? Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao

Tools from Complex Analysis

Theorem (Maximum Principle)

Let be a domain, and let f be holomorphic on . (A) |f (z)| cannot attain its maximum inside unless f is constant. (B) The real part of f cannot attain its maximum inside unless f is a constant.

Theorem (Jensen's Inequality)

Suppose f is holomorphic on the whole complex plane and f (0) = 1. Let Mf (R) = max|z|=R |f (z)|. Let Nf (t) be the number of zeros of f with norm t where a zero of multiplicity n is counted n times. Then

R 0

Nf (t) dt t

log

Mf (R).

? Relates growth of a holomorphic function to distribution of its zeroes ? Used to bound the number of zeroes of an entire function

? Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao

Theorem (Borel-Carath?odory Lemma)

Suppose f = u + iv is holomorphic on the whole complex plane. Suppose u A on

B(0, R). Then

|f (n)(0)|

2n! (A

- u(0))

Rn

? Bounds all derivatives of f at using only the real part of f

? Used in proof of Hadamard Factorization Theorem to prove that function is a polynomial by taking limit and showing that nth derivative approaches

? Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao

6

Entire Functions

Definition (Order)

The order of an entire function, f , is the infimum of all possible > 0 such that there exists constants A and B that satisfy

|f (z)| AeB|z|

? sin z, cos z have order

?

cos

z

has

order

1/2

? Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao

Entire Functions

Theorem

Let f be an entire function of order with f (0) = 1. Then, for any > 0 there exists a constant, C, that satisfies

Nf (R) CR+

Theorem

Let f be an entire function of order with f (0) = 1 and a1, a2, ... be the zeroes of f in non-decreasing order of norms. Then, for any > 0,

1

<

n=1 |an|+

In other words, the convergence index of the zeros is at most .

For cos

exzahmapsleo,rsdienrz1a/2n,dacnods

z have order , and their zeroes its zeroes grow quadratically.

grow

linearly

while

? Ishita Goluguri, Toyesh Jayaswal, Andrew Lee, Mentor: Chengyang Shao

8

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