Correlations of Sums of Two Squares and Other Arithmetic Functions in Function Fields

Abstract

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.