I am reading the Equivalents of Riemann Hypothesis: Arithmetic Equivalents pg. 35 Lemma 2.10. If $\rho$ is a non trivial zero of the Riemann Zeta function then, $\sum_{\rho}\frac{1}{\rho}=1+\frac{\gamma}{2}-\frac{1}{2} log \ 4\pi$
I want to know what is $\sum_{\rho}\frac{1}{\rho^3}$?. Please provide a reference if possible..
Edit I thought it was $\sum \rho^{-2}$, for $\sum \rho^{-3}$ it works the same way
$$\lim_{a\to \infty}\int_{a-i\infty}^{a+i\infty} \frac{\zeta'(s)}{s^3\zeta(s)}ds=0$$ This is also equal to
$$\int_{2-i\infty\to 2+i\infty\to -\infty\to 2-i\infty} \frac{\zeta'(s)}{s^3\zeta(s)}ds$$ Then apply the residue theorem, you'll get your series plus the residue at $1$, the residue at $0$ and the sum of the residues at the trivial zeros which is $\sum_{n=1}^\infty \frac1{(-2n)^3}$
The residue at $0$ depends on known constants plus $\zeta'(0),\zeta''(0),\zeta'''(0)$, by the functional equation those relate to the Laurent coefficients of $\zeta(s)$ at $s=1$