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

Abstract

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

Incorporated referee comments. Accepted for publication in Math. Annalen

References in corpus (7)

• Arithmetic functions in short intervals and the symmetric group
• The equidistribution of L-functions of twists by Witt vector Dirichlet characters over function fields
• Higher moments of arithmetic functions in short intervals: a geometric perspective
• A representation theory approach to integral moments of L-functions over function fields
• On the Chowla and twin primes conjectures over ${\mathbb{F}}_q[T]$
• Square-root cancellation for sums of factorization functions over short intervals in function fields
• Cubic twin prime polynomials are counted by a modular form

Cited by in corpus (1)

• Correlations of multiplicative functions in function fields