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

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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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Notes

  1. Adapted to our averaging method, Tong’s constant is \(c_k = \frac{2^{2-1/k}-1}{(4k-2)\pi ^2} \sum _{n=1}^\infty \frac{ d_k(n)^2}{n^{1+\frac{1}{k}}}\).

  2. The statement of [28, Theorem A], which deals with all moments, includes a term which is not small for the second moment; however the actual proof, see [28, Equation 9.8 and below] does give a good remainder.

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Authors and Affiliations

  1. School of Mathematics, University of Bristol, Bristol, BS8 1TW, UK

    J. P. Keating & E. Roditty-Gershon

  2. Department of Mathematics, University of Michigan, 530 Church St., Ann Arbor, MI, 48109, USA

    B. Rodgers

  3. Raymond and Beverly Sackler School of Mathematical Sciences, Tel Aviv University, 69978, Tel Aviv, Israel

    Z. Rudnick

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  1. J. P. Keating
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  2. B. Rodgers
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  3. E. Roditty-Gershon
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  4. Z. Rudnick
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Correspondence to J. P. Keating.

Additional information

JPK gratefully acknowledges support under EPSRC Programme Grant EP/K034383/1 LMF: L-Functions and Modular Forms, a grant from Leverhulme Trust, a Royal Society Wolfson Merit Award, a Royal Society Leverhulme Senior Research Fellowship, and by the Air Force Office of Scientific Research, Air Force Material Command, USAF, under grant number FA8655-10-1-3088. ZR is similarly grateful for support from the European Research Council under the European Union’s Seventh Framework Programme (FP7/2007–2013)/ERC Grant Agreement No. 320755, and from the Israel Science Foundation (Grant No. 925/14).

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Keating, J.P., Rodgers, B., Roditty-Gershon, E. et al. Sums of divisor functions in \(\mathbb {F}_q[t]\) and matrix integrals. Math. Z. 288, 167–198 (2018). https://doi.org/10.1007/s00209-017-1884-1

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  • DOI: https://doi.org/10.1007/s00209-017-1884-1

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