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 $\zeta^\prime(s)$. 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.