# Questions related to the Dirichlet series for $\frac{\zeta'(s)}{\zeta(s)^2}$

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?

• 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). – reuns Jan 16 at 1:04
• 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... – daniel Jan 16 at 8:30