Level curves for Zhang's Eta Function

Abstract

Study of the level curve for the real part of $\eta (s)=0$ with $\eta (s)={\pi }^{-s/2}\Gamma (s/2){\zeta }^{\prime }(s)$ gives a new classification of the zeros of $\zeta (s)$ and of ${\zeta }^{\prime }(s)$. We conjecture that for type 2 zeros, $\mathrm{liminf}({\beta }^{\prime }-1/2)\log {\gamma }^{\prime }=0$ if and only if $\mathrm{liminf}({\gamma }^{+}-{\gamma }^{-})\log {\gamma }^{\prime }=0$, and reduce the conjecture to a lower bound on the curvature of the level curve. We compute and classify $10^6$ zeros of ${\zeta }^{\prime }(s)$ near $T=10^{10}$. The Riemann Hypothesis is assumed throughout. An appendix develops the analogous classification for characteristic polynomials of unitary matrices.

Supersedes arXiv:2009.11886. To appear in Experimental Mathematics