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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.


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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 $\frac{\zeta'(s)}{\zeta(s)^2}$.


Illustration of formula (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.


Illustration of absolute part of formula (2) for s=1+it

Figure (3): Illustration of formula (2) for $\left|\frac{\zeta'(1+i\,t)}{\zeta(1+i\,t)^2}\right|$


Illustration of real part of formula (2) for s=1+it

Figure (4): Illustration of formula (2) for $\Re\left(\frac{\zeta'(1+i\,t)}{\zeta(1+i\,t)^2}\right)$


Illustration of imaginary part of formula (2) for s=1+it

Figure (5): Illustration of formula (2) for $\Im\left(\frac{\zeta'(1+i\,t)}{\zeta(1+i\,t)^2}\right)$


Illustration of argument part of formula (2) for s=1+it

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?

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    $\begingroup$ See those kind of proof of the PNT. Going from $b(n)=\mu(n)$ to $b(n) \log n$ is one of the main tools of ANT. The PNT and explicit formula for $\sum_{n \le x}\mu(n)$ are not very different to those for $\pi(x), \psi(x)$. You are supposed to understand how the residue theorem applied to the inverse Mellin transform gives a maybe non-convergent explicit formula for $\sum_{n \le x}\mu(n),\sum_{n \le x}\mu(n)\log n$. The convergence is a matter of the density of zeros and Hadamard 3 circles (see Titchmarsh). $\endgroup$ – reuns Jan 16 at 1:04
  • $\begingroup$ The comment above seems like a good answer to the questions. You would probably get good specific answers if you asked one question at a time... $\endgroup$ – daniel Jan 16 at 8:30

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