Lehmer Pairs Revisited

Abstract

ABSTRACTWe seek to understand how the technical definition of a Lehmer pair can be related to more analytic properties of the Riemann zeta function, particularly the location of the zeros of ζ′(s). Because we are interested in the connection [Csordas et al. 94] 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 PΞ′(γ) in terms of ζ′(ρ) where ρ = 1/2 + iγ. Theorem 3 expresses PΞ′(γ+) + PΞ′(γ−) in terms of nearby zeros ρ′ of ζ′(s). We examine 114, 661 pairs of zeros of ζ(s) around height t = 106, finding 855 strong Lehmer pairs. These are compared to the corresponding zeros of ζ′(s) in the same range.