# How is the Riemann-Siegel formula applied?

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:-(

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This and this have nice discussions... – J. M. Mar 28 '12 at 21:09
See this and this as well. – J. M. Mar 28 '12 at 21:13
@J.M. Wow, thanks. This might take me a while, but keep on posting! – draks ... Mar 28 '12 at 21:16
I don't have the time or the mood to write a full discussion, so I left them as comments. Somebody else with more time and motivation might want to do so in the meantime; having said that, all the essential ingredients for implementing Riemann-Siegel in your computing environment of choice are in those papers I linked to. – J. M. Mar 28 '12 at 21:20
@Aryabhata: I had initially guessed "numerically", based on some of OP's previous questions, but as it is currently written, I agree that this one's a bit open-ended... – J. M. Mar 28 '12 at 22:06