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

Sums of divisor functions in $$\mathbb {F}_q[t]$$ F q [ t ] and matrix integrals We study the mean square of sums of the kth divisor function $$d_k(n)$$ 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 $$q\rightarrow \infty $$ 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 $$d_k(n)$$ 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. http://www.deepdyve.com/assets/images/DeepDyve-Logo-lg.png Mathematische Zeitschrift Springer Journals

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

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Publisher
Springer Berlin Heidelberg
Copyright
Copyright © 2017 by Springer-Verlag Berlin Heidelberg
Subject
Mathematics; Mathematics, general
ISSN
0025-5874
eISSN
1432-1823
D.O.I.
10.1007/s00209-017-1884-1
Publisher site
See Article on Publisher Site

Abstract

We study the mean square of sums of the kth divisor function $$d_k(n)$$ 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 $$q\rightarrow \infty $$ 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 $$d_k(n)$$ 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.

Journal

Mathematische ZeitschriftSpringer Journals

Published: Mar 28, 2017

References

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