Chebotarev density theorem in short intervals for extensions of $\mathbb {F}_q(T)$
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- by Lior Bary-Soroker, Ofir Gorodetsky, Taelin Karidi and Will Sawin PDF
- Trans. Amer. Math. Soc. 373 (2020), 597-628 Request permission
Abstract:
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 $\mathbb {Q}$ with Galois group $G$, a conjugacy class $C$ in $G$, and a $1\geq \varepsilon >0$, one wants to compute the asymptotic of the number of primes $x\leq p\leq x+x^{\varepsilon }$ with Frobenius conjugacy class in $E$ equal to $C$. The level of difficulty grows as $\varepsilon$ becomes smaller. Assuming the Generalized Riemann Hypothesis, one can merely reach the regime $1\geq \varepsilon >1/2$. We establish a function field analogue of the Chebotarev theorem in short intervals for any $\varepsilon >0$. Our result is valid in the limit when the size of the finite field tends to $\infty$ 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.References
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Additional Information
- Lior Bary-Soroker
- Affiliation: Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
- MR Author ID: 797213
- ORCID: 0000-0002-1303-247X
- Email: barylior@post.tau.ac.il
- Ofir Gorodetsky
- Affiliation: Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, Tel Aviv 69978, Israel
- MR Author ID: 1234845
- Email: ofir.goro@gmail.com
- Taelin Karidi
- Affiliation: Department of Mathematics, California Institute of Technology, Pasadena, California 91125
- Email: tkaridi@caltech.edu
- Will Sawin
- Affiliation: Department of Mathematics, Columbia University, New York, New York 10027
- MR Author ID: 1022068
- Email: sawin@math.columbia.edu
- Received by editor(s): November 21, 2018
- Received by editor(s) in revised form: July 1, 2019
- Published electronically: October 1, 2019
- Additional Notes: 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. - © Copyright 2019 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 373 (2020), 597-628
- MSC (2010): Primary 11T55; Secondary 11N05, 11S20
- DOI: https://doi.org/10.1090/tran/7945
- MathSciNet review: 4042886