# What is the convergence of the explicit formula for $\frac{\zeta'(s)}{\zeta(s)}$?

The explicit formula for $$\frac{\zeta'(s)}{\zeta(s)}$$ illustrated in (3) below was derived from the relationship illustrated in (1) below using the explicit formula for $$\psi(x)$$ defined in (2) below.

(1) $$\quad\frac{\zeta'(x)}{\zeta(s)}=\frac{s}{1-s}-s\int\limits_1^{\infty}x^{-s-1}(\psi(x)-x)\,dx\,,\quad \Re(s)\ge 1\quad(\Re(s)>\frac{1}{2}\text{ assuming RH})$$

(2) $$\quad\psi_o(x)=x-\sum\limits_\rho\frac{x^{\,\rho}}{\rho}-\log(2\,\pi)-\sum\limits_{n=1}^\infty\frac{x^{-2\,n}}{-2\,n}\,,\quad x>1$$

(3) $$\quad\frac{\zeta'(s)}{\zeta(s)}=\frac{s}{1-s}+\log(2\,\pi)-\frac{1}{2}H_{\frac{s}{2}}+s\sum\limits_\rho\frac{1}{\rho\,\left(s-\rho\right)}$$

My understanding is the validity of relationship (1) above for $$\Re(s)\ge 1$$ is predicted by the Prime Number Theorem, and the validity of relationship (1) above for $$\Re(s)>\frac{1}{2}$$ is predicted by the Riemann Hypothesis (RH). Contrary to these theoretical convergences, formula (3) above seems to exhibit observational evidence of convergence everywhere I've evaluated it. Some example plots are illustrated following the two questions below.

Question (1): What is the convergence of the explicit formula for $$\frac{\zeta'(s)}{\zeta(s)}$$ illustrated in (3) above?

It's interesting to contrast the formula for $$\frac{\zeta'(s)}{\zeta(s)}$$ defined in (3) above with the following formula for $$\frac{\zeta'(s)}{\zeta(s)}$$. Note (3) above is derived from (2) above which is derived from (4) below.

(4) $$\quad\frac{\zeta'(s)}{\zeta(s)}=\frac{1}{1-s}+\frac{1}{2}\log(\pi)-\frac{1}{2}\psi^{(0)}\left(\frac{s}{2}+1\right)+\sum\limits_\rho\frac{1}{s-\rho}$$

Question (2): Can formula (3) be derived from formula (4) in a different manner?

The following plot illustrates formula (3) for $$\frac{\zeta'(s)}{\zeta(s)}$$ (orange curve) and the reference function (underlying blue curve) evaluated for $$s\in\mathbb{R}$$ where formula (3) is evaluated over the first 100 pairs of non-trivial zeta zeros.

Figure (1): Illustration of formula (3) for $$\frac{\zeta'(s)}{\zeta(s)}$$ (orange curve) evalualated for $$s\in\mathbb{R}$$

The following two plots illustrate the real and imaginary parts respectively of formula (3) for $$\frac{\zeta'(s)}{\zeta(s)}$$ (orange curves) and the corresponding parts of the reference function (underlying blue curves) evaluated along the critical line $$s=\frac{1}{2}+i\,t$$ where formula (3) is evaluated over the first 100 pairs of non-trivial zeta zeros.

Figure (2): Illustration of the real part of formula (3) for $$\frac{\zeta'(s)}{\zeta(s)}$$ (orange curve) evaluated along the critical line $$s=\frac{1}{2}+i\,t$$

Figure (3): Illustration of the imaginary part of formula (3) for $$\frac{\zeta'(s)}{\zeta(s)}$$ (orange curve) evaluated along the critical line $$s=\frac{1}{2}+i\,t$$

I don't believe the vertical orange lines at the location of the non-trivial zeta zeros illustrated in Figure (3) above indicate a lack of convergence of formula (3), rather I believe the vertical lines are a result of Mathematica connecting two adjacent evaluation points where the first evaluation point is slightly to the left of a non-trivial zeta zero and the next evaluation point is slightly to the right of the non-trivial zeta zero. Formula (3) above clearly illustrates a pole at each non-trivial zeta zero. The following table illustrates the imaginary part of formula (3) for $$\frac{\zeta'(s)}{\zeta(s)}$$ evaluates fairly closely to the imaginary part of the reference function when evaluated at $$\rho_1\pm\epsilon$$ where formula (3) was evaluated over the first 100 pairs of non-trivial zeta zeros.

(5) $$\quad\begin{array}{ccccc} \epsilon & \Im\left(\frac{\zeta'(\rho_1-\epsilon)}{\zeta(\rho_1-\epsilon)}\right) & \Im\left(\frac{\zeta'(\rho_1+\epsilon)}{\zeta(\rho _1+\epsilon)}\right) & \Im(\text{(3)}@(\rho_1-\epsilon)) & \Im(\text{(3)@}(\rho_1+\epsilon)) \\ 0.1 & 9.91199 & -10.0762 & 9.82459 & -10.1649 \\ 0.01 & 99.9173 & -100.082 & 99.8293 & -100.17 \\ 0.001 & 999.918 & -1000.08 & 999.83 & -1000.17 \\ 0.0001 & 9999.92 & -10000.1 & 9999.83 & -10000.2 \\ 0.00001 & 99999.9 & -100000. & 99999.8 & -100000. \\ \text{1.\grave{ }*{}^{\wedge}-6} & 1.\times 10^6 & -1.\times 10^6 & 1.\times 10^6 & -1.\times 10^6 \\ \text{1.\grave{ }*{}^{\wedge}-7} & 1.\times 10^7 & -1.\times 10^7 & 1.\times 10^7 & -1.\times 10^7 \\ \text{1.\grave{ }*{}^{\wedge}-8} & 1.\times 10^8 & -1.\times 10^8 & 1.\times 10^8 & -1.\times 10^8 \\ \text{1.\grave{ }*{}^{\wedge}-9} & 9.99995\times 10^8 & -1.\times 10^9 & 1.\times 10^9 & -1.\times 10^9 \\ \text{1.\grave{ }*{}^{\wedge}-10} & 1.00001\times 10^{10} & -9.9997\times 10^9 & 1.\times 10^{10} & -1.\times 10^{10} \\ \end{array}$$

• $\rho_n \approx 1/2+i C n /\ln (n)$ so it converges absolutely for every $s$ not a zero en.wikipedia.org/wiki/Z_function#Behavior_of_the_Z-function – reuns Oct 27 '18 at 0:11
• @reuns Thanks for the feedback. The question here is related to a broader investigation which is the subject of my follow-on question at math.stackexchange.com/q/2988957. I'd appreciate any feedback you might have on my new question which I suspect you'll find to be more interesting than most of my previous questions. – Steven Clark Nov 7 '18 at 19:07