# Is there a valid explicit formula for $f(x)=\sum\limits_{n=1}^x \frac{1}{n}\sum\limits_{d|n} \mu(d)\,d$?

This question is related to the function $$f(x)$$ defined in (1) below where A023900(n) is the Dirichlet inverse of Euler totient function $$\phi(n)$$. I believe the related Dirichlet series illustrated in (2) below is valid for $$\Re(s)>\frac{1}{2}$$ assuming the Riemann Hypothesis (RH).

(1) $$\quad f(x)=\sum\limits_{n=1}^x a(n),\qquad a(n)=\frac{1}{n}\sum\limits_{d|n} \mu(d)\,d=\frac{A023900(n)}{n}$$

(2) $$\quad\frac{\zeta(s+1)}{\zeta(s)}=s\int\limits_0^\infty f(x)\,x^{-s-1}\,dx=\sum\limits_{n=1}^\infty a(n)\,n^{-s},\quad\Re(s)>\frac{1}{2}\quad\text{(assuming RH)}$$

Question: Is there a valid explicit formula for the function $$f(x)$$ defined in (1) above?

The answer posted by reuns below leads to the following explicit formula for $$f(x)$$. I find the growth asymptotic $$-2\log(x)$$ and the appearance of the $$\gamma$$ term both somewhat interesting.

(3) $$\quad f(x)=-2\log(x)+2(\log(2 \pi)-\gamma)+\sum\limits_{\rho}\frac{x^{\rho}\,\zeta(\rho+1)}{\rho\,\zeta'(\rho)}+\sum\limits_{n}\frac{x^{-2 n}\,\zeta(1-2 n)}{-2 n\, \zeta'(-2 n)},\quad x>1$$

The following figure illustrates the explicit formula for $$f(x)$$ defined in formula (3) above in orange overlaid on the reference function defined in formula (1) above in blue where formula (3) is evaluated over the first $$100$$ pairs of non-trivial zeta zeros and $$30$$ trivial zeta zeros.

Figure (1): Illustration of Explicit Formula for $$f(x)$$ (orange) overlaid on reference function (blue)

It works exactly the same way as for $$1/\zeta(s)$$. There is a sequence $$T_k\to \infty$$ such that (for $$x>0$$ not an integer) $$\sum_{n\le x} (\sum_{d| n}\mu(d)\frac{1}{n/d})=1_{x >1} \lim_{k\to \infty}\sum_{|\Im(\rho)|\le T_k} Res(\frac{\zeta(s+1)}{\zeta(s)}\frac{x^s}{s},\rho)$$