Vol. 10, No. 2, 2016

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Squarefree polynomials and Möbius values in short intervals and arithmetic progressions

Jonathan P. Keating and Zeev Rudnick

Vol. 10 (2016), No. 2, 375–420
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.

Keywords
squarefrees, Möbius function, short intervals, Good–Churchhouse conjecture, Chowla's conjecture, function fields, equidistribution
Mathematical Subject Classification 2010
Primary: 11T55
Secondary: 11M38, 11M50
Milestones
Received: 15 February 2015
Revised: 9 October 2015
Accepted: 30 November 2015
Published: 16 March 2016
Authors
 Jonathan P. Keating School of Mathematics University of Bristol Bristol BS8 1TW United Kingdom Zeev Rudnick Raymond and Beverly Sackler School of Mathematical Sciences Tel Aviv University Tel Aviv 69978 Israel