Primes in arithmetic progressions to large moduli and refinements of Harman's sieve

TL;DR

This paper advances understanding of prime distribution in arithmetic progressions for large moduli by refining sieve methods and establishing new mean value theorems, leading to improved bounds on prime counts in residue classes.

Contribution

It introduces novel variants of Harman's sieve to handle larger moduli and derives new bounds for prime counts in arithmetic progressions.

Findings

Mean value theorems for primes with bilinear moduli up to x^{9/17}

Mean value theorems for primes with trilinear moduli up to x^{17/32}

New bounds for π(x; q, a) for almost all moduli q

Abstract

We study the average distribution of primes of size $x$ in arithmetic progressions to moduli larger than $x^{\frac{1}{2}}$. Using arithmetic information from the works of many authors together with different variants of the original Harman's sieve, we construct suitable majorants and minorants for the prime indicator function $\mathbb{1}_{p}(n)$ that satisfy Bombieri--Vinogradov type mean value theorems with different types of moduli. Specifically, we obtain some mean value theorems for primes with bilinear forms of moduli up to $x^{\frac{9}{17}}$ or with trilinear forms of moduli up to $x^{\frac{17}{32}}$. As a by-product, we obtain new upper and lower bounds for $\pi(x; q, a)$ that hold for almost all moduli $q$.