Some multidimensional integrals in number theory and connections with the Painlevé 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 Painleve 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. Published by AIP Publishing.