Riemann Zeta Function

Question

Can somebody provide me with the formula for the sum of reciprocal of the roots of the Riemann zeta function?

Also if $a+ib$ is a root, will $a-ib$ also be a root?

Answer 1

Lemma: If $f(z)$ is an analytic power series with real coefficients, then $f(\bar{z})=\overline{f(z)}$ for all $z\in{\bf C}$.

Try to prove this. Generalize to meromorphic $f$. Alternatively, $n^{-\bar{s}}=\overline{n^{-s}}$. Thus $\zeta(\bar{s})=\overline{\zeta(s)}$, so nontrivial zeros of the Riemann zeta function come in conjugate pairs $\{\rho,\overline{\rho}\}$.

The sum of reciprocals of nontrivial zeros of the zeta function is given by

$$\sum_\rho\frac{1}{\rho}=1+\frac{1}{2}\gamma-\frac{1}{2}\log(4\pi)$$

where $\gamma$ is the Euler-Mascheroni constant. This is unconditional on the truth of the Riemann hypothesis, but the sum is not absolutely convergent (the summands must be paired as conjugates in order to attain convergence). The sums $\sum_\rho\rho^{-k}$ for some other $k$ are also known.

See equations & discussion in $(3)$ through $(11)$ on MathWorld's article on $\zeta(s)$'s zeros.