This question is related to the following two functions evaluated with the coefficient function $a(n)=\mu(n)\log(n)$.
(1) $\quad f(x)=\sum\limits_{n=1}^x a(n)$
(2) $\quad\frac{\zeta'(s)}{\zeta(s)^2}=\sum\limits_{n=1}^\infty a(n)\,n^{-s},\quad\Re(s)>1?$
The following plot illustrates $f(x)$ defined in formula (1) above.
Figure (1): Illustration of $f(x)$ defined in formula (1)
Question (1): Is it true $f(x)$ has an infinite number of zero crossings?
Question (2): What are the limits on $f(x)$ predicted by the Prime Number Theorem and the Riemann Hypothesis?
The following figure illustrates the Dirichlet series for $\frac{\zeta'(s)}{\zeta(s)^2}$ defined in (2) above in orange where formula (2) is evaluated over the first $10,000$ terms. The underlying blue reference function is $\frac{\zeta'(s)}{\zeta(s)^2}$.
Figure (2): Illustration of formula (2) for $\frac{\zeta'(s)}{\zeta(s)^2}$ (orange curve) and reference function (blue curve)
The following four figures illustrate formula (2) for $\frac{\zeta'(s)}{\zeta(s)^2}$ evaluated along the line $\Re(s)=1$ in orange where formula (2) is evaluated over the first $1,000$ terms. The underlying blue reference function is $\frac{\zeta'(s)}{\zeta(s)^2}$. The red discrete portions of the plots illustrate the evaluation of formula (2) for $\frac{\zeta'(1+i\,t)}{\zeta(1+i\,t)^2}$ where $t$ equals the imaginary part of a non-trivial zeta zero.
Figure (3): Illustration of formula (2) for $\left|\frac{\zeta'(1+i\,t)}{\zeta(1+i\,t)^2}\right|$
Figure (4): Illustration of formula (2) for $\Re\left(\frac{\zeta'(1+i\,t)}{\zeta(1+i\,t)^2}\right)$
Figure (5): Illustration of formula (2) for $\Im\left(\frac{\zeta'(1+i\,t)}{\zeta(1+i\,t)^2}\right)$
Figure (6): Illustration of formula (2) for $Arg\left(\frac{\zeta'(1+i\,t)}{\zeta(1+i\,t)^2}\right)$
Question (3): What is the range of convergence of the Dirichlet series for $\frac{\zeta'(s)}{\zeta(s)^2}$ defined in (2) above? Does it converge only for $\Re(s)>1$, or does it also converge for $\Re(s)=1\land\Im(s)\ne 0$?
Question (4): Are there explicit formulas for $f(x)$ and $\frac{\zeta'(s)}{\zeta(s)^2}$ expressed in terms of the non-trivial zeta zeros?