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

TL;DR

This paper presents a modified linear sieve with enhanced distribution properties, leading to improved bounds on twin primes and simplifying previous complex proofs in prime number theory.

Contribution

It introduces a new sieve modification with strong factorization properties, surpassing previous distribution levels and simplifying existing proofs for twin prime counts.

Findings

Achieves equidistribution of primes up to size x in arithmetic progressions to moduli up to x^{10/17}

Provides a new upper bound on the count of twin primes

Simplifies the 2004 Wu argument and improves previous bounds significantly

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.