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

Hirotaka Kobayashi

Abstract

We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions $\frac{1}{2} + i(a n + b)$. 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 $k$.

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

Abstract

We obtain asymptotic formulae for the second discrete moments of the Riemann zeta function over arithmetic progressions . It reveals noticeable relation between the discrete moments and the continuous moment of the Riemann zeta function. Especially, when 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 for positive integers .
Paper Structure (6 sections, 2 theorems, 81 equations)

This paper contains 6 sections, 2 theorems, 81 equations.

Table of Contents

  1. Introduction
  2. the proof of theorem 1.1
  3. An estimate of $S_1$
  4. A contribution of $S_2$
  5. Conclusion
  6. the proof of theorem 1.2

Key Result

Theorem 1.1

Let $a$ be a real number such that $e^{2\pi k/a}$ is irrational for all positive integer $k$. We have, as $T \to \infty$, Moreover, when $a$ is a positive integer with $a = o((\log \log T)^{\varepsilon})$, we have for any fixed $A > 0$.

Theorems & Definitions (5)

  • Remark 1
  • Remark 2
  • Theorem 1.1
  • Remark 3
  • Theorem 1.2