Lehmer Pairs Revisited
Jeffrey Stopple · Experimental Mathematics · 2016
We 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 [Csordas et al. 94] G. Csordas, W. Smith, R. Varga. “Lehmer Pairs of Zeros, the de Bruijn-Newman Constant, and the Riemann Hypothesis.” Constr. Approx. 10 (1994), 107–129.[Crossref], [Web of Science ®] , [Google Scholar]] 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.