Asymptotic identities for additive convolutions of sums of divisors

TL;DR

This paper provides a new proof of Ramanujan's conjecture on the asymptotic behavior of additive convolutions of sum-of-divisors functions, including lower order terms and connections to geometric topology.

Contribution

It introduces a novel proof of Ramanujan's conjecture that extends the asymptotic analysis to include lower order terms for most parameter ranges.

Findings

Established asymptotic formulas with lower order terms.

Extended validity of Ramanujan's conjecture to broader parameters.

Connected additive convolutions to geometric topology problems.

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.