Investigation of the Characteristics of the Zeros of the Riemann Zeta Function in the Critical Strip Using Implicit Function Properties of the Real and Imaginary Components of the Dirichlet Eta function.

EasyChair Preprint 1312, version 1

Abstract

This paper investigates the characteristics of the zeros of the Riemann zeta function (of s) in the critical strip by using the Dirichlet eta function, which has the same zeros. The characteristics of the implicit functions for the real and imaginary components when those components are equal are investigated and it is shown that the function describing the value of the real component when the real and imaginary components are equal has a derivative that does not change sign along any of its individual curves - meaning that each value of the imaginary part of s produces at most one zero. Combined with the fact that the zeros of the Riemann xi function are also the zeros of the zeta function and xi(s) = xi(1-s), this leads to the conclusion that the Riemann Hypothesis is true.

Keyphrases: Critical Strip, Dirichlet eta, Harmonic Addition Theorem, Partial sums, Riemann Zeta, Riemann xi, analysis, derivative, implicit functions, zeros

@booklet{EasyChair:1312,
  author    = {Andrew Logan},
  title     = {Investigation of the Characteristics of the Zeros of the Riemann Zeta Function in the Critical Strip Using Implicit Function Properties of the Real and Imaginary Components of the Dirichlet Eta function.},
  howpublished = {EasyChair Preprint 1312},
  year      = {EasyChair, 2019}}