Asymptotic identities for additive convolutions of sums of divisors

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.