Odd moments in the distribution of primes

Kuperberg, Vivian

doi:10.2140/ant.2025.19.617

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Odd moments in the distribution of primes

Vivian Kuperberg

Vol. 19 (2025), No. 4, 617–666

DOI: 10.2140/ant.2025.19.617

Abstract

Montgomery and Soundararajan showed that the distribution of $ψ (x + H) - ψ (x)$ , for $0 \leq x \leq N$ , is approximately normal with mean $\sim H$ and variance $\sim H \log (N ∕ H)$ , when $N^{δ} \leq H \leq N^{1 - δ}$ . Their work depends on showing that sums $R_{k} (h)$ of $k$ -term singular series are $μ_{k} {(- h \log h + A h)}^{k ∕ 2} + O_{k} (h^{k ∕ 2 - 1 ∕ (7 k) + 𝜀})$ , where $A$ is a constant and $μ_{k}$ are the Gaussian moment constants. We study lower-order terms in the size of these moments. We conjecture that when $k$ is odd, $R_{k} (h) ≍ h^{(k - 1) ∕ 2} {(\log h)}^{(k + 1) ∕ 2}$ . We prove an upper bound with the correct power of $h$ when $k = 3$ , and prove analogous upper bounds in the function field setting when $k = 3$ and $k = 5$ . We provide further evidence for this conjecture in the form of numerical computations.

Keywords

sums of singular series, distribution of primes

Mathematical Subject Classification

Primary: 11N05, 11N13

Milestones

Received: 8 October 2021

Revised: 14 May 2024

Accepted: 15 June 2024

Published: 24 March 2025

Authors

Vivian Kuperberg
Departement Mathematik ETH Zürich Zürich Switzerland

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Vol. 19, No. 4, 2025