On the average value of the first $ n $ values of the sum-of-divisors function

Abstract

We prove that the number of positive integers $ n \leq x $ dividing $ \sigma (1) + \cdots + \sigma (n)$ is less than $x/(\log x)^{0.15742} $ for all sufficiently large $ x $, where $ \sigma $ stands for the sum-of-divisors function.