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

TL;DR

This paper explores the connection between certain number theory integrals, piecewise polynomial functions, and the Painlevé V equation, revealing their smoothness and asymptotic behaviors.

Contribution

It expresses these polynomials as inverse Fourier transforms of Hankel determinants satisfying Painlevé V, and analyzes their smoothness and asymptotics.

Findings

Polynomials are very smooth at transition points

Asymptotics of polynomials are determined near k=c/2

Connections established between number theory integrals and Painlevé V

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.