1. Riemann Zeta Function and Prime Number Theorem Korea Science Academy 08-047 Park, Min Jae

2. 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

3. History of Prime Number Theorem

4. Distribution of Primes Prime Counting Function

5. Calculating PCF Representation of PCF (C. P. Willan, 1964) Using Willson’s Theorem Many other representations

6. Heuristics Sieve of Eratosthenes

7. Heuristics Approximation Using Taylor Series

8. Approximation of PCF (Gauss, 1863) (Legendre, 1798)

9. Approximation of PCF Graph Showing Estimations

10. Prime Number Theorem Using L’Hospital’s Theorem or

11. Prime Number Theorem n’th Prime

12. Background on Complex Analysis

13. Differentiation Real-Valued Function 3 Cases of Complex Function Cauchy-Riemann Equation

14. Integration Definite Integral Real Function Complex Function

15. Integration Indefinite Integral Real Function Complex Function Require Other Conditions

16. 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.

17. Integration Cauchy’s Integral Theorem 2

18. 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,

19. Laurent Series The generalization of Taylor series. where

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

21. 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.

22. Proof of PNT with Zeta Function

23. Key Idea Chebyshev’s Weighted PCF Equivalence

24. Lemmas Lemma 1 For any arithmetical function a(n), let where A(x) = 0 if x < 1. Then

25. 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

26. 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

27. Lemmas Lemma 3 If c > 0 and u > 0, then for every positive integer k we have the integral being absolutely convergent.

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

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