1
vote
0answers
19 views

An inequality involving Möbius function [duplicate]

For any positive integer $n$ show the inequality holds : $$\left|\sum_{i=1}^{n}\frac{\mu(i)}{i}\right|\le 1$$ I tried induction. when $\mu(n+1)=0$ it is trivial. But what if $\mu(n+1)\ne 0$? I am ...
5
votes
2answers
177 views

Sum of $n \sigma(n)$

What is known about the asymptotic behavior of $$ -\frac{\pi^2}{18}x^3+\sum_{n\le x}n\sigma(n) ? $$ It seems to be $O(x^{2+\varepsilon})$ but I cannot prove this.