On the Brun--Titchmarsh theorem. II
Ping Xi, Junren Zheng
TL;DR
This paper advances the Brun--Titchmarsh bounds for primes in arithmetic progressions when the modulus $q$ is a large prime with $q\sim x^{\varpi}$. Focusing on the challenging regime near the square-root barrier, the authors obtain explicit, improved constants $C(\varpi)$ in the upper bound $\max_{(a,q)=1}\pi(x;q,a) \le (C(\varpi)+\varepsilon) \dfrac{x}{\varphi(q)\log x}$, notably achieving $C(\varpi)=\frac{240}{184-217\varpi}$ for $\varpi\in[\tfrac12,\tfrac{34}{67})$ and the endpoint $C(\tfrac12)=\frac{480}{151}$, which improves upon Iwaniec's classical results. The proof pivots on reducing the sieve remainder to quintilinear sums of Kloosterman sums and then triumphing over the inherent short-mean cancellations by employing a novel shifting trick (inspired by the Vinogradov–Burgess–Karatsuba lineage and refined by subsequent works) combined with Hölder’s inequality and the deep algebro-geometric bounds of Kowalski–Michel–Sawin on sums of products of Kloosterman sums. This synthesis leverages additive-structure exploitation to surpass the square-root barrier in certain parameter ranges, yielding sharper constants and extending the regime where Brun–Titchmarsh-type bounds are effective. The work has potential implications for zero-free regions, distribution of primes in progressions, and interplay between sieve methods and exponential-sum techniques in analytic number theory.
Abstract
Denote by $π(x;q,a)$ the number of primes $p\leqslant x$ with $p\equiv a\bmod q.$ We prove new upper bounds for $π(x;q,a)$ 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.