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I've just discovered the Abel-Plana formula: http://en.wikipedia.org/wiki/Argument_principle

I'm trying to use it to get a closed-form expression for $\sum_{n=1}^\infty \frac 1{n^3}$.

So far, I have the $$ \sum_{n=1}^\infty \frac 1{n^3}= 1+2i\int_0^{\infty} \frac{dt}{\bigl(\exp(2\pi t)-1\bigr)(it+1)^3}. $$ This integral has poles at $t=0$ and $t=i$. I know I should use the residue theorem but I'm not sure how to apply it to this integral. Any thoughts would be appreciated.

PS- I know that the answer should be $\zeta(3)$ but I want to know what the Abel-Plana formula has to say about it.

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Thanks for Te(fi)xing that up for me :) I am studying for an exam right now and just thought of it on the fly. –  joe Nov 16 '12 at 7:53
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1 Answer 1

$ζ(3)=\iiint_{[0,1]^3}\frac{1}{1-xyz}dxdydz$, up to now, the best result is that $ζ(3)$ is an irrational number. you can consider your result in the view of complex analysis.

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But how would I evaluate this integral using the Residue theorem? –  joe Nov 16 '12 at 20:05
    
That's interesting. I never saw that expression for zeta(3) –  joe Nov 20 '12 at 20:14
    
This does not answer the question. –  Potato Jul 19 '13 at 20:32
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