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Mathematics > Number Theory

arXiv:1805.08811 (math)
[Submitted on 22 May 2018 (v1), last revised 8 Dec 2019 (this version, v2)]

Title:Some multidimensional integrals in number theory and connections with the Painlevé V equation

Authors:Estelle Basor, Fan Ge, Michael O. Rubinstein
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Abstract:We study piecewise polynomial functions $\gamma_k(c)$ that appear in the asymptotics of averages of the divisor sum in short intervals. Specifically, we express these polynomials as the inverse Fourier transform of a Hankel determinant that satisfies a Painlevé V equation. We prove that $\gamma_k(c)$ is very smooth at its transition points, and also determine the asymptotics of $\gamma_k(c)$ in a large neighbourhood of $k=c/2$. Finally, we consider the coefficients that appear in the asymptotics of elliptic Aliquot cycles.
Subjects: Number Theory (math.NT)
Cite as: arXiv:1805.08811 [math.NT]
  (or arXiv:1805.08811v2 [math.NT] for this version)
  https://doi.org/10.48550/arXiv.1805.08811
Related DOI: https://doi.org/10.1063/1.5038658

Submission history

From: Michael Rubinstein [view email]
[v1] Tue, 22 May 2018 18:41:44 UTC (16 KB)
[v2] Sun, 8 Dec 2019 11:46:29 UTC (16 KB)
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