X-Ray of Zhang's eta function

TL;DR

This paper investigates the level curves of the real and imaginary parts of a modified eta function to classify zeros of the Riemann zeta function and its derivative, providing numerical and theoretical insights into their distribution.

Contribution

It introduces a new classification scheme for zeros of ta(s) related to the Riemann zeta function and proves a conjecture for a specific zero type under the Riemann Hypothesis.

Findings

Numerical evidence links zero gaps to classification types

Full conjecture of Soundararajan proved for type 2 zeros

Provides a new geometric perspective on zero distribution

Abstract

Study of the level curves the real part of $\eta(s)=0$ and imaginary part of $\eta(s)=0$, for $\eta(s)=\pi^{-s/2}\Gamma(s/2)\zeta^\prime(s)$ gives a new classification of the zeros of $\zeta(s)$ and of $\zeta^\prime(s)$. Numerical evidence indicates that the statistics of the gaps (between zeros of $\zeta$), or distance from the critical line (for zeros of $\zeta^\prime$) is related to the classification. Theorem 6 gives the full conjecture of Soundararajan for the zeros we classify as type 2. We assume the Riemann Hypothesis throughout.