Product Equation for Riemann Zeta Function

Theorem
There exists a constant $B$ such that:


 * $\ds \frac {\map {\zeta'} s} {\map \zeta s} = B - \frac 1 {s - 1} + \frac 1 2 \ln \pi - \frac 1 2 \frac {\map {\Gamma'} {s / 2 + 1} } {\map \Gamma {s / 2 + 1} } + \sum_\rho \paren {\frac 1 {s - \rho} + \frac 1 \rho}$

where:
 * $\zeta$ is the Riemann zeta function
 * $\rho$ runs over the nontrivial zeros of $\zeta$
 * $\Gamma$ is the gamma function.

Proof
Let $\xi$ be the completed Riemann zeta function:

We have that the Completed Riemann Zeta Function has Order One.

So, by the Hadamard Factorisation Theorem, there exist constants $A$, $B$ such that:


 * $\ds \map \xi s = \map \exp {A + B s} : \prod_\rho \paren {1 - \frac s \rho} \map \exp {\frac s \rho}$

where $\rho$ runs over the zeros of $\xi$, that is, the nontrivial zeros of $\zeta$.

Therefore:

Combining $(1)$ and $(2)$ we have: