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Riemann Zeta Function and Prime Number Theorem

Riemann Zeta Function and Prime Number Theorem. Korea Science Academy 08-047 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.

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Riemann Zeta Function and Prime Number Theorem

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  1. Riemann Zeta Functionand 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 • 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 • Laurent Series The generalization of Taylor series. where

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

  21. Analytic Continuation • 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 Ψ1(x)/x² • Theorem 1 If c > 1 and x ≥ 1 wehave

  29. Integral Representation for Ψ1(x)/x² • Theorem 2 If c > 1 and x ≥ 1 wehave where

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