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.

Illustration of Explicit Formula for f(x) (orange)

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


1 Answer 1


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) $$\sum_{n\le x} (\sum_{d| n}\mu(d)\frac{1}{n/d})=1_{x >1} \lim_{k\to \infty}\sum_{|\Im(\rho)|\le T_k} Res(\frac{\zeta(s+1)}{\zeta(s)}\frac{x^s}{s},\rho) $$


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