Mean-square values of the Riemann zeta function on arithmetic progressions

2023/12/18 by Hirotaka Kobayashi, Kobayashi, Hirotaka
Mathematics · #Analytic Number Theory Research #History and Theory of Mathematics #Meromorphic and Entire Functions

paper · pdf · doi:10.48550/arxiv.2401.01892

Abstract

We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions (1)/(2) + i(a n + b). It reveals noticeable relation between the discrete moments and the continuous moment of the Riemann zeta function. Especially, when a is a positive integer, main terms of the formula are equal to those for the continuous mean value. The proof requires the rational approximation of e^πk/a for positive integers k.

Related

• A mean-square value of the Riemann zeta function over an arithmetic progression
• The Riemann-zeta function on vertical arithmetic progressions
• Discrete Moments of the Riemann Zeta-Function near σ=1
• Basic analytic number theory
• Discrete Ω-results for the Riemann zeta function