Riemann Zeta Functionand Prime Number Theo-

remKorea Science Academy

08-047 Park, Min Jae

Contents

• History of Prime Number Theorem

• Background on Complex Ana-lysis

• Riemann Zeta Function• Proof of PNT with Zeta Func-

tion• Other Issues on Zeta Func-

tion• Generalization and Applica-

tion

History of Prime Number Theorem

Distribution of Primes

• Prime Counting Function

Calculating PCF

• Representation of PCF (C. P. Willan, 1964)

• Using Willson’s Theorem• Many other representations

Heuristics

• Sieve of Eratosthenes

Heuristics

• Approximation

• Using Taylor Series

Approximation of PCF

• (Gauss, 1863)

• (Legendre, 1798)

Approximation of PCF

• Graph Showing Estimations

Prime Number Theorem

• Prime Number Theorem

• Using L’Hospital’s Theorem

or

Prime Number Theorem

• n’th Prime

Background on Complex Analysis

Differentiation

• Real-Valued Function

• 3 Cases of Complex Function• Cauchy-Riemann Equation

Integration

• Definite Integral• Real Function

• Complex Function

Integration

• Indefinite Integral• Real Function

• Complex Function• Require Other Conditions

Integration

• Cauchy’s Integral TheoremIf f(z) is a function that is analytic on a simply connected region Δ, then

is a constant for every path of integration C of the region Δ.

Integration

• Cauchy’s Integral Theorem 2

Integration

• Cauchy’s Integral FormulaIf f(z) is a function that is analytic on a simply connected region Δ, then for every point z in Δ and every simple closed path of integration C,

Laurent Series

• Laurent SeriesThe generalization of Taylor series.

where

Integration

• Cauchy’s Residue TheoremLet f(z) be analytic except for isolated poles zr in a region Δ . Then

Analytic Continuation

• Analytic ContinuationIf two analytic functions are defined in a region Δ and are equivalent for all points on some curve C in Δ, then they are equivalent for all points in the region Δ.

Proof of PNT with Zeta Function

Key Idea

• Chebyshev’s Weighted PCF

• Equivalence

Lemmas

• Lemma 1For any arithmetical function a(n), let

where A(x) = 0 if x < 1. Then

Lemmas

• Abel’s IdentityFor any arithmetical function a(n), let

where A(x) = 0 if x < 1. Assume f has a continuous derivative on the interval [y, x], where 0 < y < x. Then we have

Lemmas

• Lemma 2Let and let . Assume also that a(n) is nonnegative for all n. If we have the asymptotic formula

for some c > 0 and L > 0, then we also have

Lemmas

• Lemma 3If c > 0 and u > 0, then for every positive integer k we have

the integral being absolutely convergent.

Integral Representation for Ψ1(x)/x²

• Theorem 1If c > 1 and x ≥ 1 we have

Integral Representation for Ψ1(x)/x²

• Theorem 2If c > 1 and x ≥ 1 we have

where