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arXiv · 2504.12692

On the Brun--Titchmarsh theorem. II

Ping Xi·Junren Zheng
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Abstract

Denote by $\pi(x;q,a)$ the number of primes $p\leqslant x$ with $p\equiv a\bmod q.$ We prove new upper bounds for $\pi(x;q,a)$ when $q$ is a large prime very close to $\sqrt{x}$, improving upon the classical work of Iwaniec (1982). The proof reduces to bounding a quintilinear sum of Kloosterman sums, to which we introduce a new shifting argument inspired by Vinogradov--Burgess--Karatsuba, going beyond the classical Fourier-analytic approach thanks to a deep algebro-geometric result of Kowalski--Michel--Sawin on sums of products of Kloosterman sums.

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Ping Xi, Junren Zheng. 2025-04-17. On the Brun--Titchmarsh theorem. II. https://doi.org/10.1093/imrn%2Frnaf337

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