Squarefree polynomials and möbius values in short intervals and arithmetic progressions

Jonathan P. Keating, Zeev Rudnick

Research output: Contribution to journalArticle (Academic Journal)peer-review

15 Citations (Scopus)
263 Downloads (Pure)

Abstract

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.

Original languageEnglish
Pages (from-to)375-420
Number of pages46
JournalAlgebra and Number Theory
Volume10
Issue number2
DOIs
Publication statusPublished - 16 Mar 2016

Keywords

  • Chowla’s conjecture
  • Equidistribution
  • Function fields
  • Good–Churchhouse conjecture
  • Möbius function
  • Short intervals
  • Squarefrees

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