Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

All Zeros of the Riemann Zeta Function in the Critical Strip are Located on the Critical Line and are Simple

Frank Stenger

Abstract

In this paper we study the function G(z) := int{0,infinity} y^{z-1}(1 + \exp(y))^{-1} dy, for z in C. We derive a functional equation that relates G(z) and G(1 - z) for all z in C, and we prove: -- That G and the Riemann Zeta function Zeta have exactly the same zeros in the critical region D := z in C: Re z in (0,1); -- All the zeros of the Riemann Zeta function located on the critical line are simple; and -- The Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line L := {z in D : Re z = 1/2}.

All Zeros of the Riemann Zeta Function in the Critical Strip are Located on the Critical Line and are Simple

Abstract

In this paper we study the function G(z) := int{0,infinity} y^{z-1}(1 + \exp(y))^{-1} dy, for z in C. We derive a functional equation that relates G(z) and G(1 - z) for all z in C, and we prove: -- That G and the Riemann Zeta function Zeta have exactly the same zeros in the critical region D := z in C: Re z in (0,1); -- All the zeros of the Riemann Zeta function located on the critical line are simple; and -- The Riemann hypothesis, i.e., that all of the zeros of G in D are located on the critical line L := {z in D : Re z = 1/2}.

Paper Structure

This paper contains 13 sections, 10 theorems, 22 equations.

Table of Contents

  1. Introduction and Summary
  2. Representations and related properties of $G$
  3. Properties of $\kappa$ , $\zeta$ and $G$
  4. Relevant Gamma function relations
  5. Bounds on $\kappa$
  6. Analyticity definition of multiplicity
  7. Functional equations for $\zeta$ and $G$
  8. Zeros of $G$ and $\zeta$ in $D$
  9. Definitions of $\kappa^\mp\,,\ G^{(m)}$ and $G^{[m]}$
  10. Restricting the domain of $\Im\,G^{(2\,m+1)}$
  11. Inequalities for $G^{(m)}(\sigma)$
  12. Schwarz reflection
  13. Proof of the Riemann Hypothesis

Key Result

Lemma 2.2

(i.) By replacing $z$ with $z/2$ in the duplication formula for the Gamma function, we get: (ii.) Both $\Gamma(1/2 + i\,x)$ and $\Gamma(1 + i\,x)$ are bounded by $\pi^{1/2}$ for all $x \in {\mathbf R}$ , by Equations (6.1.30) and (6.1.31) of [15]; and (iii.) The function $1/\Gamma(z)$ is an entire function; $\Gamma$ is analytic in ${\mathbf C}$ except for simple poles at $z = -\,n$ ($n = 0

Theorems & Definitions (16)

  • Definition 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Definition 2.4
  • Lemma 2.5
  • Lemma 2.6
  • Definition 2.7
  • Lemma 2.8
  • Lemma 2.9
  • Lemma 2.10
  • ...and 6 more