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

Abstract

We study piecewise polynomial functions γk(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 γk(c) is very smooth at its transition points and also determine the asymptotics of γk(c) in a large neighbourhood of k = c/2. Finally, we consider the coefficients that appear in the asymptotics of elliptic aliquot cycles.

Locations

• Journal of Mathematical Physics - View
• arXiv (Cornell University) - View - PDF
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