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The general methodology used in this answer will handle your sum: Mean Value of a Multiplicative Function close to $n$ in Terms of the Zeta Function. (See also this MSE answer, and this answer on Math Overflow) The main theorem proven in that answer is that if we let $g$ be defined so that $(1*g)(n)=\frac{f(n)}{n}$, then for any $0<\sigma<1$ we have ...


I'm assuming that that the series on the RHS is $$\sum_{n\geq1}\frac{H_{n}+\exp\left(H_{n}\right)\log\left(H_{n}\right)}{n^{s}}. $$ It's well known that $$H_{n}=\log\left(n\right)+\gamma+o\left(1\right) $$ where $\gamma $ is the Euler-Mascheroni constant. Hence the series is absolutely convergent if $\textrm{Re}\left(s\right)>2 $. About the second ...

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