Open Access
Translator Disclaimer
2016 Squarefree polynomials and Möbius values in short intervals and arithmetic progressions
Jonathan Keating, Zeev Rudnick
Algebra Number Theory 10(2): 375-420 (2016). DOI: 10.2140/ant.2016.10.375


We calculate the mean and variance of sums of the Möbius function μ and the indicator function of the squarefrees μ2, in both short intervals and arithmetic progressions, in the context of the ring Fq[t] of polynomials over a finite field Fq of q elements, in the limit q . 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.


Download Citation

Jonathan Keating. Zeev Rudnick. "Squarefree polynomials and Möbius values in short intervals and arithmetic progressions." Algebra Number Theory 10 (2) 375 - 420, 2016.


Received: 15 February 2015; Revised: 9 October 2015; Accepted: 30 November 2015; Published: 2016
First available in Project Euclid: 16 November 2017

zbMATH: 06561468
MathSciNet: MR3477745
Digital Object Identifier: 10.2140/ant.2016.10.375

Primary: 11T55
Secondary: 11M38 , 11M50

Keywords: Chowla's conjecture , equidistribution , function fields , Good–Churchhouse conjecture , Möbius function , short intervals , squarefrees

Rights: Copyright © 2016 Mathematical Sciences Publishers


Vol.10 • No. 2 • 2016
Back to Top