Riemann Zeta-Function

Author: Zhiming Ou

Publisher:    3265 Public Way

ISBN: 978-1-997036-01-2

Summary:

In this book, I will develop all expressions for the Riemann Zeta function, include the infinite series, integral, approximate, interpolation and infinite product. The integral expression is well known since E.C. Titchmarsh, my breakthrough is the integer interpolations for functions with symmetry. For an entire function with order not exceeding the order of the sine function, if it vanishes at all integers, then it can only be a constant multiple of the sine function.

In Titchmarsh’s book <The theory of the Riemann Zeta-Function> (revised by D. R. Heath-Brown), approximate formulae were given. I write them in another form, and use the Magic Numbers to estimate the partial sums of the Riemann Zeta function. This results in the proof of the Lindelof Conjecture.

Today, if Georg Friedrich Bernahard Riemann (1826-1866) were resurrected, he would be happy: not 500 years yet, his hypothesis was solved.

 

Content

Chapter 1 Introduction to Dirichlet Series

Definition

Convergence

Euler Product

Lagurre-Polya Class

Coefficient Inversion

 

Chapter 2 Prerequisite Numbers and Functions                10

Binomial Coefficients

  Stirling Numbers

Bernoulli Numbers

Euler’s Summation Formula

    Values of Riemann Zeta function at Integers

Gamma and Beta Functions

 

Chapter 3 Integral and Power Series              30

Chapter 4 Integer Interpolation                   50

Chapter 5 Approximate Formula for Riemann Zeta Function           60

Chapter 6 Final Factored Form        80