Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

An investigation of the non-trivial zeros of the Riemann zeta function

Yuri Heymann

Abstract

While many zeros of the Riemann zeta function are located on the critical line $\Re(s)=1/2$, the non-existence of zeros in the remaining part of the critical strip $\Re(s) \in \, ]0, 1[$ is the main scope to be proven for the Riemann hypothesis. The Riemann zeta functional leads to a relation between the zeros on either sides of the critical line. Given $s$ a complex number and $\bar{s}$ its complex conjugate, if $s$ is a zero of the Riemann zeta function in the critical strip $\Re(s) \in \, ]0, 1[$, then $ζ(s) = ζ(1-\bar{s})$, as a key proposition to prove the Riemann hypothesis.

An investigation of the non-trivial zeros of the Riemann zeta function

Abstract

While many zeros of the Riemann zeta function are located on the critical line , the non-existence of zeros in the remaining part of the critical strip is the main scope to be proven for the Riemann hypothesis. The Riemann zeta functional leads to a relation between the zeros on either sides of the critical line. Given a complex number and its complex conjugate, if is a zero of the Riemann zeta function in the critical strip , then , as a key proposition to prove the Riemann hypothesis.

Paper Structure

This paper contains 8 sections, 47 equations.

Table of Contents

  1. Introduction
  2. Some elementary propositions
  3. A recap on the zeros of the Riemann zeta function
  4. Poles of the Riemann zeta function
  5. Zeros when $\Re(s) > 1$
  6. Zeros when $\Re(s) <0$
  7. Zeros on the line $\Re(s)=0,1$
  8. Zeros in the critical strip $\Re(s) \in \, ]0,1[$

Theorems & Definitions (7)

  • proof
  • proof
  • proof
  • proof
  • proof
  • proof
  • proof