An Ω-theorem for an error term related to the sum-of-divisors function

Abstract

Let σ denote the sum-of-divisors function, and set$E_1 (x): = \sum\limits_{n = x} {\sigma (n) - \frac{{\pi ^2 }}{{12}}x^2 }$. Gronwall and Wigert proved (independently) in 1913 and 1914, respectively, thatE ₁ (x)=Ω (x log logx). In this paper we obtain the more preciseE ₁ (x)=Ω₋(x log logx). The method consists in averaging$E_{ - 1} (x): = \sum\limits_{n = x} {\frac{{\sigma (n)}}{n} - \left( {\frac{{\pi ^2 }}{6}x - \tfrac{1}{2}\log x} \right)}$ over suitable arithmetic progressions, and was suggested by the work ofP. Erdös andH. N. Shapiro [Canad. J. Math. 3–4, 375–385 (1951)] on the error term corresponding to Euler's functions,$\sum\limits_{n = x} {\varphi (n) - \frac{3}{{\pi ^2 }}x^2 }$.