On the exact location of the non-trivial zeros of Riemann's zeta function

TL;DR

This paper introduces a real-valued function related to the zeros of the Riemann zeta function and shows how its solutions correspond to the zeros' locations, assuming the Riemann hypothesis and simple zeros.

Contribution

It defines a new real analytic function kappa(t) that encodes the zeros of zeta(s) and establishes a direct relation between solutions of kappa(t)=n and the zeros' positions without unproved hypotheses.

Findings

kappa(t) is closely related to the zeros of zeta(s) and zeta'(s)

Under RH and simplicity, zeros satisfy kappa(t)=n uniquely

Provides a new perspective on locating zeta zeros

Abstract

In this paper we introduce the real valued real analytic function kappa(t) implicitly defined by exp(2 pi i kappa(t)) = -exp(-2 i theta(t)) * (zeta'(1/2-it)/zeta'(1/2+it)) and kappa(0)=-1/2. (where theta(t) is the function appearing in the known formula zeta(1/2+it)= Z(t) * e^{-i theta(t)}). By studying the equation kappa(t) = n (without making any unproved hypotheses), we will show that (and how) this function is closely related to the (exact) position of the zeros of Riemann's zeta(s) and zeta'(s). Assuming the Riemann hypothesis and the simplicity of the zeros of zeta(s), it will follow that the ordinate of the zero 1/2 + i gamma_n of zeta(s) will be the unique solution to the equation kappa(t) = n.