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arXiv · 1805.08811

Some multidimensional integrals in number theory and connections with the Painlev\'e V equation

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 Painlev\'e 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.

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Estelle Basor, Fan Ge, Michael O. Rubinstein. 2018-05-22. Some multidimensional integrals in number theory and connections with the Painlev\'e V equation. https://doi.org/10.1063/1.5038658

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