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Mathematics > Number Theory

arXiv:1504.03444 (math)
[Submitted on 14 Apr 2015 (v1), last revised 9 Oct 2015 (this version, v2)]

Title:Squarefree polynomials and Mobius values in short intervals and arithmetic progressions

Authors:J.P. Keating, Z. Rudnick
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Abstract:We calculate the mean and variance of sums of the Möbius 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.
Comments: Added references and corrected misprints
Subjects: Number Theory (math.NT)
Cite as: arXiv:1504.03444 [math.NT]
  (or arXiv:1504.03444v2 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.1504.03444
Journal reference: Algebra Number Theory 10 (2016) 375-420
Related DOI: https://doi.org/10.2140/ant.2016.10.375

Submission history

From: Zeev Rudnick [view email]
[v1] Tue, 14 Apr 2015 07:49:51 UTC (35 KB)
[v2] Fri, 9 Oct 2015 12:43:49 UTC (35 KB)
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