I am trying to reproduce the following bound:
$\sum_{1\leq m\leq x, m\neq n}\frac{1}{|\log(m/n)|}=O(x\log(x))$,
for $x\geq 2$ and some $n$, $1\leq n\leq x$ (the implicit constant shouldn't depend on $n$).
I have tried to bound each term by $\frac{1}{|\log(1\pm 1/x)|}$, but this doesn't seem to be good enough. Splitting the sum at $\log(x)$ also didn't help. Any ideas?
Thank you very much for your comments or hints.