, Volume 103, Issue 2, pp 145-157

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

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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, \(\sum\limits_{n = x} {\varphi (n) - \frac{3}{{\pi ^2 }}x^2 }\) .