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.
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Research Article| July 10 2018
Some multidimensional integrals in number theory and connections with the Painlevé V equation
Special Collection: In Memory of Ludwig Faddeev
Estelle Basor, Fan Ge, Michael O. Rubinstein; Some multidimensional integrals in number theory and connections with the Painlevé V equation. J. Math. Phys. 1 September 2018; 59 (9): 091404. https://doi.org/10.1063/1.5038658
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