Work

Simple Barban–Davenport–Halberstam type asymptotics for general sequences

Year: 2025

Type: article

Abstract: We prove two estimates for the Barban–Davenport–Halberstam type variance of a general complex sequence in arithmetic progressions. The proofs are elementary, and our estimates are capable of yielding an asymptotic for the variance when the sequence is sufficiently nice, and is either somewhat sparse or is sufficiently like the integers in its divisibility by small moduli. As a concrete application, we deduce a Barban–Davenport–Halberstam type variance asymptotic for the ‐smooth numbers less than , on a wide range of the parameters. This addresses a question considered by Granville and Vaughan. Our methods also recover some earlier results for the case of prime numbers.

Source: Journal of the London Mathematical Society

DOI: 10.1112/jlms.70293

Author: Adam J. Harper

Institution: University of Warwick

Language: English

Topic: Analytic Number Theory Research

Subfield: Algebra and Number Theory

Field: Mathematics

Domain: Physical Sciences

Open Access status: hybrid

Estimated APC paid: $3,930

Funders: Centre de Recherches Mathématiques, Vetenskapsrådet, Engineering and Physical Sciences Research Council