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The second moment of the Riemann zeta function at its local extrema

Christopher Hughes, Solomon Lugmayer, Andrew Pearce-Crump

Abstract

Conrey and Ghosh studied the second moment of the Riemann zeta function, evaluated at its local extrema along the critical line, finding the leading order behaviour to be $\frac{e^2 - 5}{2 π} T (\log T)^2$. This problem is closely related to a mixed moment of the Riemann zeta function and its derivative. We present a new approach which will uncover the lower order terms for the second moment as a descending chain of powers of logarithms in the asymptotic expansion.

The second moment of the Riemann zeta function at its local extrema

Abstract

Conrey and Ghosh studied the second moment of the Riemann zeta function, evaluated at its local extrema along the critical line, finding the leading order behaviour to be . This problem is closely related to a mixed moment of the Riemann zeta function and its derivative. We present a new approach which will uncover the lower order terms for the second moment as a descending chain of powers of logarithms in the asymptotic expansion.

Paper Structure

This paper contains 9 sections, 10 theorems, 108 equations, 2 figures, 2 tables.

Table of Contents

  1. Background
  2. Results
  3. Overview of the paper
  4. Preliminary Lemmas
  5. Proof of Theorem \ref{['thm:LowerOrderTerms']}
  6. Setting up the proof
  7. Initial manipulations of the Cauchy integral
  8. Calculating the main terms
  9. Graphical Data

Key Result

Theorem 1

Assume the Riemann Hypothesis. Given $\gamma$, an ordinate of a non-trivial zero of $\zeta (s)$, let $\gamma^+\geq \gamma$ denote the next successive ordinate of a zeta zero. Let $L = \log \frac{T}{2\pi}$. Then for any fixed $N\geq 0$, as $T\to\infty$ we have where and for $n \geq 1$ where the $c_{k,\ell}$ are the Laurent series coefficients around $s=1$ of

Figures (2)

  • Figure 1: The blue curve shows the true value of $\sum_{0< \gamma \leq T} \max_{\gamma \leq t \leq \gamma^+} \left| \zeta \left( \frac{1}{2} + it \right) \right|^2$. The green curve shows the leading order asymptotic $\frac{e^2 - 5}{4 \pi} T (\log \frac{T}{2\pi})^2$. The range of the graph is over the first million zeros of zeta.
  • Figure 2: The error between the true value and the asymptotic value given in Theorem \ref{['thm:LowerOrderTerms']} for various $N$, plotted over the first million zeros of zeta

Theorems & Definitions (22)

  • Theorem 1
  • Remark
  • Lemma 1
  • Lemma 2
  • proof
  • Lemma 3
  • Lemma 4
  • Remark
  • proof
  • Lemma 5
  • ...and 12 more