What is the application of the Riemann-Siegel formula:
$$ \zeta(s) = \sum_{n=1}^N\frac{1}{n^s} + \gamma(1-s)\sum_{n=1}^M\frac{1}{n^{1-s}} + R(s) , $$ where $ \displaystyle\gamma(s) = \pi^{1/2-s}\Gamma(s/2)/\Gamma((1-s)/2) $ is the factor appearing in the functional equation $\zeta(s) = \gamma(s) \zeta(1 − s)$, and $$ R(s) = \frac{-\Gamma(1-s)}{2\pi i}\int \frac{(-x)^{s-1}e^{-Nx}dx}{e^x-1} $$ is a contour integral whose contour starts and ends at +∞ and circles the singularities of absolute value at most 2πM.
It is mentioned here, but I don't see its usage. The Mathworld site doesn't help at all:-(
