Sum of the sum-of-divisors function I was looking at the abstract of a paper 1 which claims that [2] and [3] prove
$$
\sum_{n\le x}\sigma(n)-\frac{\pi^2}{12}x^2=\Omega(x\log\log x).
$$
But I cannot find the above—or indeed, anything approaching it—in [2]. Have I missed something?
The paper [3] clearly discusses the appropriate function and presumably gives the indicated result. I must decipher its notation, though: the author seems to use $\sigma(n)$ to denote what would usually be written $\sigma_{-1}(n)=\sigma(n)/n.$
References
1 Y.-F. S. Pétermann, "An Ω-theorem for an error term related to the sum-of-divisors function", Monatshefte für Mathematik 103:2 (1987), pp. 145-157. 
[2] T. H. Gronwall, "Some asymptotic expressions in the theory of numbers", Trans. Amer. Math. Soc. 14 (1913), pp. 113–122. JSTOR
[3] S. Wigert, Sur quelques fonctions arithmétiques, Acta Math. 37 (1914), pp. 113–140.
 A: The clue to understanding the relevance of the quoted results seems to be given in Remark 2 of Pétermann's paper (at the very end). Where it is said that a result on the limes superior of $\sigma_{-1}(x)/ \log \log x$ implies an Omega-result on the error term $E_{-1}$ .  (And thus $E_{1}$ which is the one in the  question.) This result mentioned in Remark 2, also actually appears in the paper of Gronawall, see Eq. (25) there; except it is staed for $\sigma_1$ , but this translates directly as commented at the beginning of that paper where the relation between $\sigma_{a}$ and $\sigma_{-a}$ is mentioned. 
ps. This was written a bit quickly, I hope I still got the details right, but in any case this Remark 2 seems  to be helpful in understanding the relation to the earlier papers.
A: This is proved in G. Tenenbaum's book (Introduction to analytic and probabilistic number theory), page 39 (section 3.3, theorem 3). I agree that Gronwall's paper, other than the fact that it studies the same function, seem to be completely unrelated.
A: It is not clear whether you are asking for a proof/reference for the displayed formula, or an evaluation of the contents of the cited papers. 
Dickson's History, Volume 1, page 323, says Wigert proved $$\sum_{n\le x}\sigma(n)={\pi^2x^2\over12}+x((1/2)\log x-\psi(x))+O(x)$$ where $$\psi(x)=x\sum_{n\gt x}{1\over n^2}+\sum_{n\le x}{1\over n}\rho\left({x\over n}\right)$$ and $\rho(x)$ is the fractional part of $x$. Further, for $x$ sufficiently large, $$((1/4)-\epsilon)\log x\lt\psi(x)\lt((3/4)+\epsilon)\log x$$ It seems to me that this gives a poorer error term than the one in your display. Dickson also says Landau gave corrections and simplifications to Wigert's proofs, Gottingsche gelehrte Anzeigen 177 (1915) 377-414. 
A: Error is known to be $O(x \log^{2/3}x)$. See Walfisz's book on exponential sums
