Lehmer pairs and derivatives of Hardy's $Z$-function

TL;DR

This paper establishes a new criterion for identifying Lehmer pairs of zeros of the Riemann zeta function using derivatives of Hardy's $Z$-function, linking zero distribution with the Riemann Hypothesis and Newman's conjecture.

Contribution

It introduces a derivative-based condition for Lehmer pairs and connects stationary points of the $Z$-function to Newman's conjecture, providing new analytical tools and numerical insights.

Findings

Derived a condition for Lehmer pairs using derivatives of Hardy's $Z$-function

Connected stationary points of the $Z$-function with Newman's conjecture

Presented numerical results supporting the theoretical findings

Abstract

Occurrences of very close zeros of the Riemann zeta function are strongly connected with Lehmer pairs and with the Riemann Hypothesis. The aim of the present note is to derive a condition for a pair of consecutive simple zeros of the $\zeta$-function to be a Lehmer pair in terms of derivatives of Hardy's $Z$-function. Furthermore, we connect Newman's conjecture with stationary points of the $Z$-function, and present some numerical results.