Squarefree polynomials and Möbius values in short intervals and arithmetic progressions

Abstract

We calculate the mean and variance of sums of the Möbius function $\mu$ and the indicator function of the squarefrees ${\mu }^2$, in both short intervals and arithmetic progressions, in the context of the ring ${\mathbb{F}}_q\left[t\right]$ of polynomials over a finite field ${\mathbb{F}}_q$ 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.