Work

Odd moments and adding fractions

Year: 2025

Type: article

Abstract: We prove near‐optimal upper bounds for the odd moments of the distribution of coprime residues in short intervals, confirming a conjecture of Montgomery and Vaughan. As an application, we prove near‐optimal upper bounds for the average of the refined singular series in the Hardy–Littlewood conjectures concerning the number of prime ‐tuples for odd. The main new ingredient is a near‐optimal upper bound for the number of solutions to when is odd, with and restrictions on the size of the numerators and denominators, which is of independent interest.

Source: Proceedings of the London Mathematical Society

DOI: 10.1112/plms.70068

Authors: Thomas F. Bloom, Vivian Kuperberg

Institutions: University of Manchester, ETH Zurich

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $4,700

OpenAlex ID: W4412119979

FWCI:

Cites: 18

Cited by:

Related to: 10