| Abstract: |
| Let $\sigma(n)$ be the sum of all divisors of $n$ and let $[t]$ be the integral part of $t$.
In this paper, we shall prove that
$$
\sum_{n\le x} \sigma\Big(\Big[\frac{x}{n}\Big]\Big)
= \frac{\pi^2}{6} x\log x + O(x(\log x)^{2/3}(\log_2x)^{4/3})
$$
for $x\to\infty$, and that the error term of this asymptotic formula is $\Omega(x)$. |
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