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

Ofir Gorodetsky, Will Sawin

Abstract

For a fixed polynomial $Δ$, we study the number of polynomials $f$ of degree $n$ over $\mathbb F_q$ such that $f$ and $f+Δ$ 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 $Δ$ in a manner which is consistent with the Hardy-Littlewood Conjecture. We obtain a saving of $q$ if we consider monic polynomials only and $Δ$ is a scalar. To do this, we use symmetries of the problem to get for free a small amount of averaging in $Δ$. 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.

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

Abstract

For a fixed polynomial , we study the number of polynomials of degree over such that and are both irreducible, an -analogue of the twin primes problem. In the large- limit, we obtain a lower-order term for this count if we consider non-monic polynomials, which depends on in a manner which is consistent with the Hardy-Littlewood Conjecture. We obtain a saving of if we consider monic polynomials only and is a scalar. To do this, we use symmetries of the problem to get for free a small amount of averaging in . This allows us to obtain additional saving from equidistribution results for -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.

Paper Structure

This paper contains 31 sections, 29 theorems, 183 equations.

Table of Contents

  1. Introduction
  2. Arithmetic functions on FqT and previous work
  3. Main results
  4. Further results
  5. Discussion
  6. Preliminaries
  7. Hayes characters
  8. Equivalence relation
  9. Representative sets
  10. Characters
  11. Structure of G(Rl,M)
  12. $L$-functions
  13. Relations between CovAq(alpha,beta;n,Delta) and CovMq(alpha,beta;n,Delta)
  14. Some Fourier expansions
  15. Proof of Theorem \ref{['maintermcovthm']}
  16. ...and 16 more sections

Key Result

Theorem 1

Let $n$ be a positive integer. Let $\Delta$ be a squarefree polynomial in $\mathcal{A}_q$ which is either of degree $\le n-5$ or of degree $n-1$. We have, for $n \ge 5$, If $\Delta \in \mathbb{F}_q^{\times}$ and $n \ge 4$ then

Theorems & Definitions (51)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Corollary 1
  • Remark 1
  • Conjecture 1
  • Theorem 4
  • Theorem 5
  • Theorem 6
  • Lemma 1
  • ...and 41 more