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Correlation of arithmetic functions over \(\mathbb {F}_q[T]\)

  • Ofir GorodetskyEmail author
  • Will Sawin
Article

Abstract

For a fixed polynomial \(\varDelta \), we study the number of polynomials f of degree n over \({\mathbb {F}}_q\) such that f and \(f+\varDelta \) are both irreducible, an \({\mathbb {F}}_q[T]\)-analogue of the twin primes problem. In the large-q limit, we obtain a lower-order term for this count if we consider non-monic polynomials, which depends on \(\varDelta \) in a manner which is consistent with the Hardy–Littlewood Conjecture. We obtain a saving of q if we consider monic polynomials only and \(\varDelta \) is a scalar. To do this, we use symmetries of the problem to get for free a small amount of averaging in \(\varDelta \). This allows us to obtain additional saving from equidistribution results for L-functions. We do all this in a combinatorial framework that applies to more general arithmetic functions than the indicator function of irreducibles, including the Möbius function and divisor functions.

Mathematics Subject Classification

11N37 11T55 

Notes

Acknowledgements

We wish to thank Lior Bary-Soroker, Ze’ev Rudnick and the anonymous referee for useful comments.

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Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2019

Authors and Affiliations

  1. 1.Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael
  2. 2.Department of MathematicsColumbia UniversityNew YorkUSA

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