Asymptotic identities for additive convolutions of sums of divisors

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.

revised to rewrite the introduction, highlight main theorems and update references