For squarefree $i$ what is $\sum_1^{n} \frac{1}{i}$?

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For squarefree $i$ what is $\sum_1^{n} \frac{1}{i}$ ? I use $\sum_{\sqrt{n}>m>1} \mu(m) ln(\frac{n}{m^2}+\frac{1}{2})$. I know about the connection with $\zeta(2)$ and $\frac{\zeta(s)}{\zeta(2s)}$ but that does not help ??

Can I do better ?