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, $\liminf (\beta^\prime -1/2)\log\gamma^\prime = 0$ if and only if $\liminf (\gamma^+-\gamma^-)\log \gamma^\prime=0$, 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.