  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.
