Lehmer pairs revisited

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 $ζ^\prime(s)$. Because we are interested in the connection between Lehmer pairs and the de Bruijn-Newman constant $Λ$, 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 $Ξ(t)$, 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 $ζ^\prime(ρ)$. Theorem 3 expresses the criteria for strong Lehmer pairs in terms of nearby zeros $ρ^\prime$ of $ζ^\prime(s)$. We examine 114661 pairs of zeros of $ζ(s)$ around height t=10^6, finding 855 strong Lehmer pairs. These are compared to the corresponding zeros of $ζ^\prime(s)$ in the same range.