On the discrete mean of the derivative of Hardy's Z-function

2018/11/12 by Hirotaka Kobayashi, Kobayashi, Hirotaka
Mathematics · #Advanced Harmonic Analysis Research #Analytic Number Theory Research #FOS: Mathematics #Holomorphic and Operator Theory #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.1811.04530

openalex publication_date 2018/11/12 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

Update: This result was obtained by Milinovich with a better error term. He used ζ'(s), but we considered Z'(t). We corrected a typo in the main theorem. We consider the sum of the square of the derivative of Hardy's Z-function over the zeros of Hardy's Z-function. If the Riemann Hypothesis is true, it is equal to the sum of |ζ'(ρ)|², where ρ runs over the zeros of the Riemann zeta-function. In 1984, Gonek obtained an asymptotic formula for the sum. In this paper we prove a sharper formula.

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