2015/08/24 by Jeffrey Stopple, Stopple, Jeffrey · 1 citation
Mathematics · #11M50 #11Y35 #Advanced Algebra and Geometry #Advanced Mathematical Identities #Analytic Number Theory Research #FOS: Mathematics #Number Theory (math.NT)
paper · pdf · doi:10.48550/arxiv.1508.05870
openalex publication_date 2015/08/24 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
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^′(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^′(ρ). Theorem 3 expresses the criteria for strong Lehmer pairs in terms of nearby zeros ρ^′ of ζ^′(s). We examine 114661 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.