Lehmer pairs revisited

TL;DR

This paper explores the relationship between Lehmer pairs, the zeros of the Riemann zeta function, and the de Bruijn-Newman constant, using analytic and computational methods to identify and analyze strong Lehmer pairs.

Contribution

It introduces the concept of strong Lehmer pairs via the pre-Schwarzian derivative and establishes their connection to zeros of ta'(), providing new criteria and computational evidence.

Findings

Identified 855 strong Lehmer pairs among 114,661 zero pairs at height 10^6.

Proved that strong Lehmer pairs are a subset of Lehmer pairs.

Connected the properties of Lehmer pairs to zeros of ta'() and the de Bruijn-Newman constant.

Abstract

We seek to understand how the technical definition of Lehmer pair can be related to more analytic properties of the Riemann zeta function, particularly the location of the zeros of $\zeta^\prime(s)$. Because we are interested in the connection between Lehmer pairs and the de Bruijn-Newman constant $\Lambda$, we assume the Riemann Hypothesis throughout. We define strong Lehmer pairs via an inequality on the derivative of the pre-Schwarzian of Riemann's function $\Xi(t)$, evaluated at consecutive zeros. Theorem 1 shows that strong Lehmer pairs are Lehmer pairs. Theorem 2 describes the derivative of the pre-Schwarzian in terms of $\zeta^\prime(\rho)$. Theorem 3 expresses the criteria for strong Lehmer pairs in terms of nearby zeros $\rho^\prime$ of $\zeta^\prime(s)$. We examine 114661 pairs of zeros of $\zeta(s)$ around height t=10^6, finding 855 strong Lehmer pairs. These are compared to the corresponding zeros of $\zeta^\prime(s)$ in the same range.