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The discrete second moment of mixed derivatives of the Riemann zeta function

Benjamin Durkan, Christopher Hughes, Andrew Pearce-Crump

TL;DR

This paper establishes a full asymptotic expansion, with power-saving or RH-conditional errors, for the discrete second moment $\sum_{0<\gamma\le T}\zeta^{(\mu)}(\rho)\zeta^{(\nu)}(1-\rho)$ of mixed derivatives of the Riemann zeta function. The authors develop a contour-integral framework, decompose the right- and left-hand vertical segments via functional equations into convergent Dirichlet series, and extract the main term as a polynomial $\mathcal{P}_{\mu,\nu}(x)$ in $\log(T/2\pi)$ by residues at $s=1$ with coefficients tied to Laurent expansions around $s=1$. Under RH they obtain $O(T^{1/2+\varepsilon})$ error; unconditionally the error is $O(T e^{-C\sqrt{\log T}})$. The results generalize prior work (Gonek, Conrey--Snaith, Milinovich) to all derivatives, recover Milinovich4s polynomial for $\mu=\nu=1$, and yield corollaries for higher derivatives and related mixed moments, with explicit coefficient data in terms of Laurent constants. This advances precise moment calculations in analytic number theory and informs conjectures about zeta-derivative statistics at zeros.

Abstract

We establish the full asymptotic for the discrete second moment of the Riemann zeta function of mixed derivatives evaluated at the zeta zeros, providing both unconditional and conditional error terms. This was first studied by Gonek, where only the leading order asymptotic was given, later extended by Conrey--Snaith and Milinovich to include the lower order terms for the first derivative. We extend the case of the first derivative to all derivatives.

The discrete second moment of mixed derivatives of the Riemann zeta function

TL;DR

This paper establishes a full asymptotic expansion, with power-saving or RH-conditional errors, for the discrete second moment of mixed derivatives of the Riemann zeta function. The authors develop a contour-integral framework, decompose the right- and left-hand vertical segments via functional equations into convergent Dirichlet series, and extract the main term as a polynomial in by residues at with coefficients tied to Laurent expansions around . Under RH they obtain error; unconditionally the error is . The results generalize prior work (Gonek, Conrey--Snaith, Milinovich) to all derivatives, recover Milinovich4s polynomial for , and yield corollaries for higher derivatives and related mixed moments, with explicit coefficient data in terms of Laurent constants. This advances precise moment calculations in analytic number theory and informs conjectures about zeta-derivative statistics at zeros.

Abstract

We establish the full asymptotic for the discrete second moment of the Riemann zeta function of mixed derivatives evaluated at the zeta zeros, providing both unconditional and conditional error terms. This was first studied by Gonek, where only the leading order asymptotic was given, later extended by Conrey--Snaith and Milinovich to include the lower order terms for the first derivative. We extend the case of the first derivative to all derivatives.
Paper Structure (14 sections, 13 theorems, 109 equations, 3 figures, 2 tables)

This paper contains 14 sections, 13 theorems, 109 equations, 3 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Statement of Results
  3. Brief outline of the proof
  4. The right-hand vertical segment
  5. Initial manipulations
  6. Evaluating the inner sum without the logarithm
  7. Reinserting the logarithm in the inner sum
  8. The left-hand vertical segment
  9. Initial manipulations
  10. Initial Manipulations of $J(\mu,\nu)$
  11. Evaluating the inner sum
  12. Proof of the Corollaries
  13. The second derivative
  14. Graphical illustration

Key Result

Theorem 1

For positive integers $\mu,\nu$, we have as $T\to\infty$, where $C$ is a positive constant and where $\mathcal{P}_{\mu,\nu}(x)$ is the polynomial of degree $\mu+\nu+2$ given by where where $c^{(\mu,k)}_{j}$ are the Laurent series coefficients around $s=1$ of and where where $d_{j}^{(\nu,k)}$ are the Laurent series coefficients around $s=1$ of If one assumes the Riemann Hypothesis, the error

Figures (3)

  • Figure 1: The first derivative
  • Figure 2: The second derivative
  • Figure 3: The mixed first and second derivatives

Theorems & Definitions (25)

  • Remark
  • Theorem 1
  • Corollary 1
  • Corollary 2
  • Corollary 3
  • Lemma 1
  • Lemma 2
  • Lemma 3
  • proof
  • Lemma 4
  • ...and 15 more