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arXiv · 2007.09275

Asymptotic identities for additive convolutions of sums of divisors

Robert J. Lemke Oliver·Sunrose T. Shrestha·Frank Thorne
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Abstract

In a 1916 paper, Ramanujan studied the additive convolution $S_{a, b}(n)$ of sum-of-divisors functions $\sigma_a(n)$ and $\sigma_b(n)$, and proved an asymptotic formula for it when $a$ and $b$ are positive odd integers. He also conjectured that his asymptotic formula should hold for all positive real $a$ and $b$. Ramanujan's conjecture was subsequently proved by Ingham, and then by Halberstam with a power saving error term. In this paper, we give a new proof of Ramanujan's conjecture that obtains lower order terms in the asymptotics for most ranges of the parameters. We also describe a connection to a counting problem in geometric topology that was studied in the second author's thesis and which served as our initial motivation in studying this sum.

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Robert J. Lemke Oliver, Sunrose T. Shrestha, Frank Thorne. 2020-07-17. Asymptotic identities for additive convolutions of sums of divisors. https://arxiv.org/abs/2007.09275

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