Sum of the sum-of-divisors function

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.

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.

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.