Sums of divisor functions in Fq[t] and matrix integrals

Jon Keating, Brad Rodgers, Edva A Roditty-Gershon, Z. Rudnick

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

11 Citations (Scopus)
203 Downloads (Pure)

Abstract

We study the mean square of sums of the kth divisor function dk(n) over short intervals and arithmetic progressions for the rational function field over a finite field of q elements. In the limit as q→∞q→∞ we establish a relationship with a matrix integral over the unitary group. Evaluating this integral enables us to compute the mean square of the sums of dk(n) in terms of a lattice point count. This lattice point count can in turn be calculated in terms of a certain piecewise polynomial function, which we analyse. Our results suggest general conjectures for the corresponding classical problems over the integers, which agree with the few cases where the answer is known.
Original languageEnglish
Pages (from-to)167-198
Number of pages32
JournalMathematische Zeitschrift
Volume288
Early online date28 Mar 2017
DOIs
Publication statusPublished - 1 Feb 2018

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