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Generalisations of the Landau--Gonek Theorem and applications to mean values of zeta

Benjamin Durkan, Christopher Hughes, Andrew Pearce-Crump

TL;DR

The work generalizes Landau–Gonek-style sums by weighting the non-trivial zeros of the Riemann zeta function with the functional equation factor $\chi(\rho)$ and the term $X^{\rho}$, establishing a uniform asymptotic for the three regimes of $X$ relative to the height $T$. The authors develop a stationary-phase-based analysis of a pivotal integral $J(\sigma,r,T)$ and a contour-analytic evaluation of the explicit formula, then leverage the approximate functional equation to control discrete moments of $\zeta$ and its derivatives. The main result, Theorem mainthm, yields explicit, region-dependent main terms and sharp error bounds, enabling rigorous proofs of Shanks' conjecture and its generalisation via mean-value arguments. This advances the understanding of how the arithmetic nature of $X$ influences zero-sum formulas and discrete moments, with implications for precise evaluations of $\zeta$-moments and zeta-derivative sums.

Abstract

The Landau--Gonek Theorem evaluates $X^ρ$ summed over the non-trivial zeros of the Riemann zeta function. Their result shows great sensitivity to the arithmetic nature of $X$. We prove a related result concerning the sum of $χ(ρ) X^ρ$ over the zeros of zeta, where $χ(s)$ is the term arising in the functional equation for the zeta function. Again, this result depends deeply on whether $X$ is an integer or not. We show the result splits into three cases, depending on whether $X$ is smaller than $T$, about the same size as $T$, or bigger than $T$. The reason this result is useful is that it easily permits the calculation of discrete moments of the Riemann zeta function via the approximate functional equation. As an application of this result, we provide an alternative proof of Shanks' conjecture.

Generalisations of the Landau--Gonek Theorem and applications to mean values of zeta

TL;DR

The work generalizes Landau–Gonek-style sums by weighting the non-trivial zeros of the Riemann zeta function with the functional equation factor and the term , establishing a uniform asymptotic for the three regimes of relative to the height . The authors develop a stationary-phase-based analysis of a pivotal integral and a contour-analytic evaluation of the explicit formula, then leverage the approximate functional equation to control discrete moments of and its derivatives. The main result, Theorem mainthm, yields explicit, region-dependent main terms and sharp error bounds, enabling rigorous proofs of Shanks' conjecture and its generalisation via mean-value arguments. This advances the understanding of how the arithmetic nature of influences zero-sum formulas and discrete moments, with implications for precise evaluations of -moments and zeta-derivative sums.

Abstract

The Landau--Gonek Theorem evaluates summed over the non-trivial zeros of the Riemann zeta function. Their result shows great sensitivity to the arithmetic nature of . We prove a related result concerning the sum of over the zeros of zeta, where is the term arising in the functional equation for the zeta function. Again, this result depends deeply on whether is an integer or not. We show the result splits into three cases, depending on whether is smaller than , about the same size as , or bigger than . The reason this result is useful is that it easily permits the calculation of discrete moments of the Riemann zeta function via the approximate functional equation. As an application of this result, we provide an alternative proof of Shanks' conjecture.
Paper Structure (6 sections, 16 theorems, 89 equations, 1 figure)

This paper contains 6 sections, 16 theorems, 89 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Results
  3. Overview of paper
  4. Proof of Theorem \ref{['mainthm']}
  5. Deduction of Shanks' conjecture
  6. Deduction of the generalised Shanks' conjecture

Key Result

Theorem 1

Given fixed $X>1$ we have as $T\to\infty$, where $\Lambda(n)$ is the von Mangoldt function.

Figures (1)

  • Figure 1: The complex values of $S(X,T)$, plotted for $T<300,000$ with $X=2000$. Note that the $\Re$- and $\Im$-axes have very different scalings

Theorems & Definitions (28)

  • Theorem 1: Landau, 1911
  • Theorem 2: Gonek, 1985
  • Corollary 2.1: Gonek, 1993
  • Theorem 3: Fujii, 1994
  • Theorem 4: Kaptan--Karabulut--Yıldırım, 2011
  • Theorem 5
  • Remark
  • Corollary 5.1
  • proof : Proof of Corollary \ref{['cor:X=1']}
  • Corollary 5.2
  • ...and 18 more