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Bilinear Kloosterman sums in function fields and the distribution of irreducible polynomials

Part of: Multiplicative number theory Finite fields and commutative rings (number-theoretic aspects)

Published online by Cambridge University Press:  20 December 2024

Christian Bagshaw
Christian Bagshaw*
Affiliation:
School of Mathematics and Statistics, University of New South Wales, Sydney
*
e-mail: [email protected]
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Abstract

Inspired by the work of Bourgain and Garaev (2013), we provide new bounds for certain weighted bilinear Kloosterman sums in polynomial rings over a finite field. As an application, we build upon and extend some results of Sawin and Shusterman (2022). These results include bounds for exponential sums weighted by the Möbius function and a level of distribution for irreducible polynomials beyond 1/2, with arbitrary composite modulus. Additionally, we can do better when averaging over the modulus, to give an analogue of the Bombieri-Vinogradov Theorem with a level of distribution even further beyond 1/2.

Keywords

Function fieldfinite fieldexponential sumKloosterman sumBombieri-Vinogradovlevel of distribution of primes

MSC classification

Primary: 11T06: Polynomials 11T23: Exponential sums 11N05: Distribution of primes
Type
Article
Information
Canadian Journal of Mathematics , First View , pp. 1 - 26
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Copyright
© The Author(s), 2024. Published by Cambridge University Press on behalf of Canadian Mathematical Society

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Footnotes

The author is very grateful to Bryce Kerr and Igor Shparlinski for many long and helpful discussions about this work, and for reading over multiple drafts of this paper. The author would also like to thank the anonymous reviewer of this paper for their feedback and suggestions. During the preparation of this work, the author was supported by an Australian Government Research Training Program (RTP) Scholarship.

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