We study piecewise polynomial functions γ 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 Painlevé V equation. We prove that γ k (c) is very smooth at its transition points, and also determine the asymptotics of γ k (c) in a large neighbourhood of k = c/2. Finally, we consider the coefficients that appear in the asymptotics of elliptic Aliquot cycles.with ∆ k (X) denoting the remainder term.1991 Mathematics Subject Classification. Primary 11M06, Secondary 33E17.
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