Riemann Zeta Function and Prime Number Theorem Korea Science Academy Park, Min Jae

Contents History of Prime Number Theorem Background on Complex Analysis Riemann Zeta Function Proof of PNT with Zeta Function Other Issues on Zeta Function Generalization and Application

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 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 Theorem If 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 Formula If 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 The generalization of Taylor series. where

Integration Cauchy’s Residue Theorem Let f(z) be analytic except for isolated poles z r in a region Δ. Then

Analytic Continuation If 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 1 For any arithmetical function a(n), let where A(x) = 0 if x < 1. Then

Lemmas Abel’s Identity For 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 2 Let 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 3 If c > 0 and u > 0, then for every positive integer k we have the integral being absolutely convergent.

Integral Representation for Ψ 1 ( x )/ x ² Theorem 1 If c > 1 and x ≥ 1 we have

Integral Representation for Ψ 1 ( x )/ x ² Theorem 2 If c > 1 and x ≥ 1 we have where