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.