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Arithmetic exponent
pairs for algebraic trace functions and applications
Jie Wu and Ping XiAppendix: Will Sawin |
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Vol. 15 (2021), No. 9, 2123–2172
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Abstract |
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We study short sums of algebraic trace functions via the -analogue of the van der Corput method, and develop theory of arithmetic exponent pairs that coincide with the classical case when the moduli have sufficiently good factorizations. As an application, we prove a quadratic analogue of the Brun–Titchmarsh theorem on average, bounding the number of primes such that . The other two applications include a larger level of distribution of divisor functions in arithmetic progressions and a sub-Weyl subconvex bound of Dirichlet -functions studied previously by Irving. |
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Keywords
$q$-analogue of the van der Corput method, arithmetic
exponent pairs, trace functions of $\ell$-adic sheaves,
Brun–Titchmarsh theorem, linear sieve
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Mathematical Subject Classification
Primary: 11L05, 11L07, 11N13, 11N36, 11T23
Secondary: 11M06, 11N37
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Milestones
Received: 30 April 2018
Revised: 27 December 2020
Accepted: 15 April 2021
Published: 23 December 2021
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Authors |
| Jie Wu | |
| School of Mathematics and
Statistics Qingdao University Qingdao China |
CNRS, UMR 8050 Université Paris-Est Créteil Paris France |
| Ping Xi | |
| School of Mathematics and
Statistics Xi’an Jiaotong University Xi’an China |
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| Will Sawin | |
| Department of Mathematics Columbia University New York, NY United States |
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