Arithmetic correlations over large finite fields

TL;DR

This paper investigates the behavior of arithmetic function correlations over large finite fields, revealing significant cancellation effects and providing insights into how these correlations diminish as the field size grows.

Contribution

It introduces methods to compute averages of lower-order terms in finite fields, detecting correlations that vanish in the large field limit, advancing understanding of arithmetic functions in function fields.

Findings

Significant cancellation in correlation averages

Correlations diminish as finite field size increases

Limitations on the size of remainder terms in asymptotic formulas

Abstract

The auto-correlations of arithmetic functions, such as the von Mangoldt function, the M\"obius function and the divisor function, are the subject of classical problems in analytic number theory. The function field analogues of these problems have recently been resolved in the limit of large finite field size $q$. However, in this limit the correlations disappear: the arithmetic functions become uncorrelated. We compute averages of terms of lower order in $q$ which detect correlations. Our results show that there is considerable cancellation in the averaging and have implications for the rate at which correlations disappear when $q \rightarrow\infty$; in particular one cannot expect remainder terms that are of the order of the square-root of the main term in this context.