A modification of the linear sieve, and the count of twin primes

Lichtman, Jared Duker

doi:10.2140/ant.2025.19.1

Download this article
	For screen
	For printing

Recent Issues

The Journal

About the journal

Ethics and policies

Peer-review process

Submission guidelines

ISSN 1944-7833 (online)

ISSN 1937-0652 (print)

Author index

To appear

Other MSP journals

A modification of the linear sieve, and the count of twin primes

Jared Duker Lichtman

Vol. 19 (2025), No. 1, 1–38

DOI: 10.2140/ant.2025.19.1

Abstract

We introduce a modification of the linear sieve whose weights satisfy strong factorization properties, and consequently equidistribute primes up to size $x$ in arithmetic progressions to moduli up to $x^{10 ∕ 17}$ . This surpasses the level of distribution $x^{4 ∕ 7}$ with the linear sieve weights from well-known work of Bombieri, Friedlander, and Iwaniec, and which was recently extended to $x^{7 ∕ 12}$ by Maynard. As an application, we obtain a new upper bound on the count of twin primes. Our method simplifies the 2004 argument of Wu, and gives the largest percentage improvement since the 1986 bound of Bombieri, Friedlander, and Iwaniec.

Keywords

linear sieve, well-factorable weights, level of distribution, switching principle, Buchstab identity

Mathematical Subject Classification

Primary: 11N35, 11N36

Secondary: 11N05

Milestones

Received: 8 May 2022

Revised: 9 January 2024

Accepted: 13 February 2024

Published: 4 December 2024

Authors

Jared Duker Lichtman
Mathematical Institute University of Oxford Oxford United Kingdom

This article is currently available only to readers at paying institutions. If enough institutions subscribe to this Subscribe to Open journal for 2025, the article will become Open Access in early 2025. Otherwise, this article (and all 2025 articles) will be available only to paid subscribers.

Vol. 19, No. 1, 2025