Squarefree polynomials and Mobius values in short intervals and arithmetic progressions

TL;DR

This paper studies the statistical behavior of the Möbius function and squarefree indicator sums over polynomials in finite fields, revealing parallels and differences with classical number theory results, using matrix integrals and equidistribution techniques.

Contribution

It provides new formulas for mean and variance of these sums in finite fields, extending the range of known results and highlighting subtle differences from integer cases.

Findings

Mean and variance formulas for sums over finite fields

Results mirror conjectures and known results for integers

Extended range of validity compared to number field cases

Abstract

We calculate the mean and variance of sums of the M\"obius function and the indicator function of the squarefrees, in both short intervals and arithmetic progressions, in the context of the ring of polynomials over a finite field of $q$ elements, in the limit $q\to \infty$. We do this by relating the sums in question to certain matrix integrals over the unitary group, using recent equidistribution results due to N. Katz, and then by evaluating these integrals. In many cases our results mirror what is either known or conjectured for the corresponding problems involving sums over the integers, which have a long history. In some cases there are subtle and surprising differences. The ranges over which our results hold is significantly greater than those established for the corresponding problems in the number field setting.