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

## 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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