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arXiv · 2602.20917

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

Runbo Li
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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 $q$.

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Runbo Li. 2026-02-24. Primes in arithmetic progressions to large moduli and refinements of Harman's sieve. https://arxiv.org/abs/2602.20917

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