# What is asymptotic and error bound for $\sum\limits_{k=1}^K\left(\frac{1}{\rho_k}+\frac{1}{\rho_{-k}}\right)$ as a function of $K$?

This is a follow-on of my previous question

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

My previous question was related to the following two formulas.

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

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

Combining formulas (1) and (2) above leads to formula (3) below.

(3) $$\quad\sum\limits_{k=1}^K\left(\frac{1}{\rho_k}+\frac{1}{\rho_{-k}}\right)=1-\frac{1}{2}\,\log(4\,\pi)+\frac{\gamma }{2}\,,\quad K\to\infty$$

Here are a few links related to formula (3) above.

Weisstein, Eric W. "Li's Criterion." From MathWorld--A Wolfram Web Resource.

Wikipedia Article on Li's Criterion

http://oeis.org/A074760

Formula (3) is illustrated for various ranges of $$K$$ following the three questions below.

Question (1): What is the asymptotic for the left side of formula (3) as a function of $$K$$?

Question (2): Does the Riemann Hypothesis place any constraints on the deviation of the left side of formula (3) from this asymptotic as a function of $$K$$?

Question (3): Is the Dirichlet series $$\sum\limits_{k=1}^\infty\left(\frac{1}{\rho_k}+\frac{1}{\rho_{-k}}\right)\,k^{-s}$$ of any theoretical importance?

The following four plots illustrate the left and right sides of formula (3) above in blue and orange respectively over several ranges of $$K$$.