Mean values of ratios of the Riemann zeta function

2023/07/16 by Daodao Yang, Yang, Daodao
Mathematics · #Advanced Mathematical Identities #Analytic Number Theory Research #Analytic and geometric function theory #FOS: Mathematics #Number Theory (math.NT)

paper · pdf · doi:10.48550/arxiv.2307.08091

openalex publication_date 2023/07/16 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/01

Abstract

It is proved that ∫_T^2T |\fracζ((1)/(2)+\rm i t)ζ(1+2\rm i t)|² \rm d t = (1)/(ζ(2)) T log T + ( \fraclog \frac2π + 2γ-1 ζ(2) -4 \fracζ^′(2)ζ²(2) ) T + O(T (log T)^-2023 ) , ∀ T \geqslant 100. For given a ∈ \mathbb N, we also establish similar formulas for second moments of |ζ(\tfrac12 + \rm i t)/ζ(1 + \rm i at)|. We have lim_{a → ∞} lim_{T → ∞}(1)/(T log T) ∫_T^2T |\fracζ((1)/(2)+\rm i t)ζ(1+\rm i at)|² \rm d t = (ζ(2))/(ζ(4)).

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