$$ \sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+O(n\log n). $$
Is the next asymptotic term known? That is, is there a monotonic increasing function $f(x)$ such that $$ \sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+f(n)+o(f(n)) $$ ? (I would guess something like $f(x)=cx$ if $f$ exists.) Alternately, are there monotonic increasing functions $f(x),g(x)$ with $$ \sum_{k=1}^n\sigma(k)=\frac{\pi^2}{12}n^2+f(n)+\Omega_\pm(g(n)) $$ ?
Apologies for what may be a basic reference question; I've misplaced by copy of Hardy & Wright and lack more advanced references like Apostol.