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Moments of the Riemann zeta function at its local extrema

Andrew Pearce-Crump

Abstract

Conrey, Ghosh and Gonek studied the first moment of the derivative of the Riemann zeta function evaluated at the non-trivial zeros of the zeta function, resolving a problem known as Shanks' conjecture. Conrey and Ghosh studied the second moment of the Riemann zeta function evaluated at its local extrema along the critical line to leading order. In this paper we combine the two results, evaluating the first moment of the zeta function and its derivatives at the local extrema of zeta along the critical line, giving a full asymptotic. We also consider the factor from the functional equation for the zeta function at these extrema.

Moments of the Riemann zeta function at its local extrema

Abstract

Conrey, Ghosh and Gonek studied the first moment of the derivative of the Riemann zeta function evaluated at the non-trivial zeros of the zeta function, resolving a problem known as Shanks' conjecture. Conrey and Ghosh studied the second moment of the Riemann zeta function evaluated at its local extrema along the critical line to leading order. In this paper we combine the two results, evaluating the first moment of the zeta function and its derivatives at the local extrema of zeta along the critical line, giving a full asymptotic. We also consider the factor from the functional equation for the zeta function at these extrema.

Paper Structure

This paper contains 13 sections, 14 theorems, 179 equations.

Table of Contents

  1. Background
  2. Results
  3. Remarks on the previous results
  4. Overview of the proofs
  5. Preliminary Lemmas
  6. Proof Theorem \ref{['thm:FirstMomnthDerivRho1']}
  7. Initial Manipulations
  8. Deriving the main terms
  9. The integral $I_1$
  10. The integral $I_2$
  11. Outline of the proof of Theorem \ref{['thm:FirstMomRho1']}
  12. Proof Theorem \ref{['thm:ChiRho1']}
  13. Outline of the proof of Theorem \ref{['thm:ChiRho']}

Key Result

Theorem 1

Assume the Riemann Hypothesis. Write $L~=~\log T/2\pi$. For $\lambda$ defined by $Z'(\lambda)=0$ and $K, n \geq 1$ positive integers, we have as $T \rightarrow \infty$ where: In these coefficients $a_m, b_m$, the $c_{\ell}^{k,j}$ are the Laurent series coefficients around $s=1$ of

Theorems & Definitions (26)

  • Theorem 1
  • Corollary 1
  • Theorem 2
  • Theorem 3
  • Remark
  • Theorem 4
  • Lemma 1
  • Lemma 2
  • proof
  • Remark
  • ...and 16 more