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Mathematische Zeitschrift

, Volume 288, Issue 1–2, pp 167–198 | Cite as

Sums of divisor functions in \(\mathbb {F}_q[t]\) and matrix integrals

  • J. P. Keating
  • B. Rodgers
  • E. Roditty-Gershon
  • Z. Rudnick
Article

Abstract

We study the mean square of sums of the kth divisor function \(d_k(n)\) over short intervals and arithmetic progressions for the rational function field over a finite field of q elements. In the limit as \(q\rightarrow \infty \) 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 \(d_k(n)\) 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.

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Copyright information

© Springer-Verlag Berlin Heidelberg 2017

Authors and Affiliations

  • J. P. Keating
    • 1
  • B. Rodgers
    • 2
  • E. Roditty-Gershon
    • 1
  • Z. Rudnick
    • 3
  1. 1.School of MathematicsUniversity of BristolBristolUK
  2. 2.Department of MathematicsUniversity of MichiganAnn ArborUSA
  3. 3.Raymond and Beverly Sackler School of Mathematical SciencesTel Aviv UniversityTel AvivIsrael

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