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Chebotarev density theorem in short intervals for extensions of F_q(T)

Bary-Soroker, Lior and Gorodetsky, Ofir and Karidi, Taelin and Sawin, Will (2020) Chebotarev density theorem in short intervals for extensions of F_q(T). Transactions of the American Mathematical Society, 373 (1). pp. 597-628. ISSN 0002-9947. doi:10.1090/tran/7945.

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An old open problem in number theory is whether the Chebotarev density theorem holds in short intervals. More precisely, given a Galois extension E of Q with Galois group G, a conjugacy class C in G, and a 1 ≥ ε > 0, one wants to compute the asymptotic of the number of primes x ≤ p ≤ x+x^ε with Frobenius conjugacy class in E equal to C. The level of difficulty grows as ε becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime 1 ≥ ε > 1/2. We establish a function field analogue of the Chebotarev theorem in short intervals for any ε > 0. Our result is valid in the limit when the size of the finite field tends to ∞ and when the extension is tamely ramified at infinity. The methods are based on a higher dimensional explicit Chebotarev theorem and applied in a much more general setting of arithmetic functions, which we name G-factorization arithmetic functions.

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Additional Information:© 2019 American Mathematical Society. Received by the editors November 21, 2018, and, in revised form, July 1, 2019. Article electronically published on October 1, 2019. The first author was partially supported by a grant of the Israel Science Foundation. Part of the work was done while the first author was a member of the Simons CRM Scholar-in-Residence Program. The research of the second author was supported by the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 320755. This research was partially conducted during the period the fourth author served as a Clay Research Fellow and partially conducted during the period he was supported by Dr. Max Rössler, the Walter Haefner Foundation, and the ETH Zurich Foundation.
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Israel Science FoundationUNSPECIFIED
Simons FoundationUNSPECIFIED
European Research Council (ERC)320755
Walter Haefner FoundationUNSPECIFIED
American Mathematical SocietyUNSPECIFIED
Issue or Number:1
Classification Code:2010 Mathematics Subject Classification: Primary 11T55; Secondary 11N05, 11S20
Record Number:CaltechAUTHORS:20200305-133205801
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Usage Policy:No commercial reproduction, distribution, display or performance rights in this work are provided.
ID Code:101726
Deposited By: Tony Diaz
Deposited On:05 Mar 2020 23:23
Last Modified:16 Nov 2021 18:05

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