On the Brun--Titchmarsh theorem. II
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arXiv
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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| _version_ | 1866909918477418496 |
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| author | Xi, Ping Zheng, Junren |
| author_facet | Xi, Ping Zheng, Junren |
| contents | 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. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2504_12692 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | On the Brun--Titchmarsh theorem. II Xi, Ping Zheng, Junren Number Theory 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. |
| title | On the Brun--Titchmarsh theorem. II |
| topic | Number Theory |
| url | https://arxiv.org/abs/2504.12692 |