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A well-known way for analytic continuing riemann zeta function is using from the functional equation between $\zeta$, $\theta$ and $\Gamma$ function. but I know that there is or there are other ways for doing this analytic continuation. what are this ways?

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migrated from mathoverflow.net Feb 16 '15 at 15:51

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  • $\begingroup$ How do you know they exist if you do not know what they are? $\endgroup$ – Tobias Kildetoft Feb 16 '15 at 11:19
  • $\begingroup$ as i have in my mind in complex analysis stein's book it says that there are some other ways for doing it. @TobiasKildetoft $\endgroup$ – user120269 Feb 16 '15 at 11:22
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    $\begingroup$ Have a look at this Wikipedia article. $\endgroup$ – abx Feb 16 '15 at 11:24
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In Titchmarsh, "The theory of the Riemann Zeta function", you can find 7 methods to prove the analytic continuation and the functional equation.

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Plenty of other ways. Take for instance the following direct computation, similar to the Euler-MacLaurin's method. Assume first $s\in \mathbb C$ with $\Re s>1$. You have $$ \zeta(s)=\sum_{n\ge 1}\frac{1}{n^s}=\sum_{n\ge 1}\left(\frac{1}{n^s}-\int_n^{n+1}\frac{dx}{x^s} \right)+\frac1{s-1}, $$ and thus $$ \zeta(s)-\frac1{s-1}=\sum_{n\ge 1}\int_n^{n+1}(n^{-s}-x^{-s}) d(x-n-1) =\sum_{n\ge 1}\int_n^{n+1}s x^{-s-1}(n+1-x)dx. $$ The rhs is obviously holomorphic for $\Re s>0$ (since $n+1-x\in [0,1]$). It is easy to go on (with more integration by parts) and prove that the rhs is in fact an entire function.

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