This question assumes the following definitions where $s\in\mathbb{C}$.
(1) $\quad f(s)=\frac{s}{s+1}$
(2) $\quad H(s)=\psi(s+1)+\gamma\qquad\text{(analytic harmonic number function)}$
Question: Can the following two relationships be proven where $\mathbb{Z}^-$ is the set of negative integers?
(3) $\quad H(s)=\sum\limits_{n=1}^\infty\frac{1}{n}f\left(\frac{s}{n}\right),\qquad s\notin\mathbb{Z}^-$
(4) $\quad f(s)=\sum\limits_{n=1}^\infty\frac{\mu(n)}{n}\ H\left({\frac{s}{n}}\right),\quad s\notin\mathbb{Z}^-$
Formula (3) above leads to the Mellin transform of $H(s)$ defined in (5) below which subsequently leads to the explicit formula for $H(s)$ defined in (6) below.
(5) $\quad\mathcal{M}_s\left[H(s)\right](z)=-\pi\ \csc(\pi z)\ \zeta(1-z)\,,\quad \Re(z)>-1$
(6) $\quad H(s)=\frac{1}{2 s}+\log(s)+\gamma+\sum\limits_{n=1}^\infty s^{-2 n}\ \zeta(1-2 n)$
After conjecturing the relationships illustrated in (3) and (4) above based on evidence obtained from plots using finite evaluation limits, I searched the following websites but couldn't seem to find any mention of these relationships. I did find an asymptotic expansion for $H(s)$ in the first two links below consistent with the explicit formula defined in (6) above, and subsequently noticed Mathematica simplifies formula (3) above to $H(s)$. These two observations seem to corroborate the correctness of relationships (3) and (4) above, but it seems to me if relationships (3) and (4) above were well known relationships they would have been mentioned somewhat prominently on the web pages at the links below because of their elegant beauty and simplicity.
Wikipedia Article: Harmonic Number
Sondow, Jonathan and Weisstein, Eric W. "Harmonic Number." From MathWorld--A Wolfram Web Resource.