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

Juan Arias de Reyna¹ ; Jan van de Lune²

¹ Facultad de Matemáticas Universidad de Sevilla Apdo. 1160 41080 Sevilla, Spain
² Langebuorren 49 9074 CH Hallum The Netherlands (formerly at the CWI, Amsterdam)

Acta Arithmetica, Tome 163 (2014) no. 3, pp. 215-245.

Résumé

We introduce the real valued real analytic function $\kappa (t)$ implicitly defined by \[ e^{2\pi i \kappa (t)}=-e^{-2i\vartheta (t)}\frac {\zeta '(1/2-it)}{\zeta '(1/2+it)} \hskip 1em (\kappa (0)=-1/2).\] By studying the equation $\kappa (t) = n$ (without making any unproved hypotheses), we 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 follows that the ordinate of the zero $1/2 + i \gamma _n$ of $\zeta (s)$ is the unique solution to the equation $\kappa (t) = n$.

Détail

DOI : 10.4064/aa163-3-3

Keywords: introduce real valued real analytic function kappa implicitly defined kappa e vartheta frac zeta it zeta hskip kappa studying equation kappa without making unproved hypotheses function closely related exact position zeros riemanns zeta zeta assuming riemann hypothesis simplicity zeros zeta follows ordinate zero gamma zeta unique solution equation kappa