On the exact location of the non-trivial zeros of Riemann's zeta function
Volume 163 / 2014
Acta Arithmetica 163 (2014), 215-245
MSC: Primary 11M26; Secondary 11M41, 26E05.
DOI: 10.4064/aa163-3-3
Abstract
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 $\kappa (t) = n$.
