Correlations of sums of two squares and other arithmetic functions in function fields

TL;DR

This paper studies correlations of sums of two squares in function fields, providing numerical evidence and proofs in large finite fields, with applications to shifted primes and higher divisor functions.

Contribution

It offers the first proof of a conjecture on autocorrelations of sums of two squares in function fields and extends methods to other arithmetic functions.

Findings

Numerical evidence supporting the conjecture

Proof of the conjecture in the large finite field limit

Applications to sums of two squares on shifted primes and higher divisor functions

Abstract

We investigate a function field analogue of a recent conjecture on autocorrelations of sums of two squares by Freiberg, Kurlberg and Rosenzweig, which generalizes an older conjecture by Connors and Keating. In particular, we provide extensive numerical evidence and prove it in the large finite field limit. Our method can also handle correlations of other arithmetic functions and we give applications to (function field analogues of) the average of sums of two squares on shifted primes, and to autocorrelations of higher divisor functions twisted by a quadratic character.