Product Equation for Riemann Zeta Function/Warning

Warning concerning Product Equation for Riemann Zeta Function

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.

Warning

The sum $\ds \sum_\rho \frac 1 {\size \rho}$ diverges, so we must be careful of the order in which we take the terms.

By the Functional Equation for Riemann Zeta Function, the zeroes occur in complex conjugate pairs, and:

$\ds \frac 1 \rho + \frac 1 {\overline \rho} = \frac {2 \map \Re \rho} {\size \rho^2} \le \frac 2 {\size \rho^2}$

and we see by the corollary to Zeroes of Functions of Finite Order that a sum of such terms does converge.