Thread: Riemann Zeta Function
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Old December 26th, 2008, 10:59 PM
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Default Riemann Zeta Function
I have a couple question regarding the Riemann zeta function and its extension onto the complex plane. I know the function is given by the reflection formula \zeta(s) = 2^s\pi^{s-1}\sin\left(\frac{\pi s}{2}\right)\Gamma(1-s)\zeta(1-s).

My first question is how is this proved?

My second question is how does this extend \zeta(s) to the complex plane?

The way I look at it is to plug in a value already known, which would be some s>1 and solve for \zeta(1-s). But this only solves for all s \in (-\infty,0) leaving \mathbb{C}\(-\infty,0) \cup [1,\infty) unaccounted for...

Thanks guys
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