  We provide uniform bounds on mean values of multiplicative functions under
very general hypotheses, detecting certain power savings missed in known
results in the literature. As an application, we study the distribution of the
sum-of-divisors function $\sigma(n)$ in coprime residue classes to moduli $q
\le (\log x)^K$, obtaining extensions of results of \'Sliwa that are uniform in
a wide range of $q$ and optimal in various parameters. As a consequence of our
results, we obtain that the values of $\sigma(n)$ sampled over $n \le x$ with
$\sigma(n)$ coprime to $q$ are asymptotically equidistributed among the coprime
residue classes mod $q$, uniformly for odd $q \le (\log x)^K$. On the other
hand, if $q$ is even, then equidistribution is restored provided we restrict to
inputs $n$ having sufficiently many prime divisors exceeding $q$.
