I was looking at [the abstract](http://rd.springer.com/article/10.1007/BF01630684) of a paper [1](https://link.springer.com/article/10.1007/BF01630684) 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](https://link.springer.com/article/10.1007/BF01630684) Y.-F. S. Pétermann, ["An Ω-theorem for an error term related to the sum-of-divisors function"](https://link.springer.com/article/10.1007/BF01630684), *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](http://www.jstor.org/stable/1988773)


[3] S. Wigert, [Sur quelques fonctions arithmétiques](http://home.us.archive.org/stream/actamathematica37upps#page/n145/mode/2up), *Acta Math.* **37** (1914), pp. 113–140.



Sum of the sum-of-divisors function
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.


