TITLE: Sum of $n \sigma(n)$
QUESTION [6 upvotes]: What is known about the asymptotic behavior of
$$
-\frac{\pi^2}{18}x^3+\sum_{n\le x}n\sigma(n) ?
$$
It seems to be $O(x^{2+\varepsilon})$ but I cannot prove this.

REPLY [3 votes]: See my blog post regarding the average of $\sigma(n)$.  This post is a two part series, part I looks at the upper bound, and Part II proves Pétermann's lower bound, which is significantly more difficult.  All results regarding $n\sigma(n)$ follow right away from partial summation.
In Part I, the hyperbola method is used to show that $$\sum_{n\leq x} \sigma(n) =\sum_{n\leq x}\sigma(n)=\frac{\pi^{2}}{12}x^2+O(x\log x),$$ which should be exactly what you are looking for.  From here, partial summation yields $$\sum_{n\leq x} n\sigma(n) =\frac{\pi^{2}}{18}x^3+O(x^2\log x).$$
I will post Part II soon which proves that the error term is not $o(x^2\log \log x),$ and oscillates from negative to positive with a magnitude of $x^2\log \log x$ infinitely often.