Could someone give me a hint on the computation of the asymptotic bound for the following series $$ \sum_{n\leq x}\frac{\log n }{ \varphi(n)}\,, $$ where $\varphi(n)$ is the Euler totient function? Also a bibliography suggestion would be ok.
2026-04-08 17:29:09.1775669349
Asymptotic bound of the series $\sum_{n\leq x}\log n / \varphi(n)$
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Hint: $$\underset{n\leq x}{\sum}\frac{1}{\varphi\left(n\right)}=C\left(\log\left(x\right)+O\left(1\right)\right)$$ where $C=\frac{315\zeta\left(3\right)}{2\pi^{4}}$ . Now use Abel summation formula http://en.wikipedia.org/wiki/Abel%27s_summation_formula.