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September 2022 On the Chowla and twin primes conjectures over $\mathbb{F}_q[T]$
Will Sawin, Mark Shusterman
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Ann. of Math. (2) 196(2): 457-506 (September 2022). DOI: 10.4007/annals.2022.196.2.1

Abstract

Using geometric methods, we improve on the function field version of the Burgess bound and show that, when restricted to certain special subspaces, the Möbius function over $\mathbb{F}_q[T]$ can be mimicked by Dirichlet characters. Combining these, we obtain a level of distribution close to $1$ for the Möbius function in arithmetic progressions and resolve Chowla's $k$-point correlation conjecture with large uniformity in the shifts. Using a function field variant of a result by Fouvry-Michel on exponential sums involving the Möbius function, we obtain a level of distribution beyond $1/2$ for irreducible polynomials, and establish the twin prime conjecture in a quantitative form. All these results hold for finite fields satisfying a simple condition.

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Will Sawin. Mark Shusterman. "On the Chowla and twin primes conjectures over $\mathbb{F}_q[T]$." Ann. of Math. (2) 196 (2) 457 - 506, September 2022. https://doi.org/10.4007/annals.2022.196.2.1

Information

Published: September 2022
First available in Project Euclid: 28 June 2022

Digital Object Identifier: 10.4007/annals.2022.196.2.1

Subjects:
Primary: 11N13 , 11P32 , 11R58 , 11T06 , 14F20

Keywords: level of distribution for irreducible polynomials , parity barrier over function fields , short character sums , twin irreducible polynomials

Rights: Copyright © 2022 Department of Mathematics, Princeton University

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Vol.196 • No. 2 • September 2022
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