An \(\Omega\)-theorem for an error term related to the sum-of-divisors functions
Let \(\sigma\) denote the sum of divisors function, and set \(E_ 1(x):=\sum_{n\leq x}\sigma (n)-(\pi^ 2/12)x^ 2.\) Gronwall and Wigert proved (independently) in 1913/14 that \(E_ 1(x)=\Omega (x \log \log x).\) In this paper we obtain the more precise \(E_ 1(x)=\Omega_-(x \log \log x).\) The method consists in averaging \[ E_{- 1}(x):=\sum_{n\leq x}(\sigma (n)/n)-((\pi^ 2/6)x-(1/2) \log x) \] over suitable arithmetic progressions, and was suggested by the work of \textit{P. Erdős} and \textit{H. N. Shapiro} [Can. J. Math. 3, 375-385 (1951; Zbl 0044.039)] on the error term corresponding to Euler's function, \(\sum_{n\leq x}\phi (n)-(3/\pi^ 2)x^ 2.\)
scientific article; zbMATH DE number 3979173