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

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 $k=c\mathrm{/}2$. Finally, we consider the coefficients that appear in the asymptotics of elliptic Aliquot cycles.