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Given the function $Z(s,N)= \sum \limits_{n=1}^{N}n^{-s}$.

In the limit $N \to \infty$ the function $Z(s,N) \to \zeta (s)$ Riemann Zeta function.

My question is: Is there a Functional equation for this function? I mean a relationship of the form $ Z(s,N)=G(s) Z(1-s,N)$.

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$\text{}{}n=0$? –  anon Dec 2 '11 at 23:54
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"... in the limit $N \to \infty$ ...": this claim is only true if $\Re(s) > 1$. –  Matt E Dec 3 '11 at 2:10
    
OH. sorry :( i meant n=1 :) otherwise the series would be divergent ... –  Jose Garcia Dec 3 '11 at 11:28
    
Edited in accord with comments above. –  Gerry Myerson Feb 27 '12 at 22:27
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2 Answers

Well, here's an answer - no, because then $G(s)=Z(s,N)Z(1-s,N)^{-1}$ would be invariant with respect to $N$, but it clearly isn't (check the graphs for various $N$ on W|A if you want to, or just plug in the trivial $N=1$ case). If you let $G$ vary with $N$ then of course there are such functions, but I doubt there is a nice formula that covers all the cases. The fact that a "$p=\infty$" factor can be added to $\zeta(s)$ to "complete" it so it obtains symmetry about $\operatorname{Re}(s)=1/2$ derives from the functional equation of the theta function (see my previous answer), and I don't think this can even have a finite analogue here because the Poisson summation formula requires a lattice to work with (of the discrete subgroup, not poset, variety), which will necessarily be infinite in the Euclidean setting.

There is one other thing I want to point out. The functional equations for $L$ functions are more suggestive in forms like $\chi(1-s)L(1-s)=\chi(s)L(s)$ because this showcases the "completed" zeta function, which may be a more "natural" object than $L$ itself, despite first appearances. As I hinted at earlier on the "$p=\infty$" factor $\pi^{-s/2}\Gamma(s/2)$, there is an underlying adelic framework that explains this sort of completion for Riemann's zeta (see this or other discussions of Tate's thesis).

Also, $Z(s,N)$ is typically denoted $H_{N,s}$ as a generalization of the harmonic numbers.


As Gerry notes, there is an approximate reflection equation given by the Riemann-Siegel formula:

$$\zeta(s)=\sum_{n=1}^N \frac{1}{n^s}+\pi^{1/2-s}\frac{\Gamma\left(\frac{s}{2}\right)}{\Gamma\left(\frac{1-s}{2}\right)}\sum_{k=1}^M\frac{1}{k^{1-s}}-\frac{\Gamma(1-s)}{2\pi i}\oint_{\gamma(2\pi M)}\frac{(-z)^{s-1}e^{-Nz}}{e^z-1}dz $$

See Wikipedia or MathWorld.

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There is something like a functional equation in Section 3 of http://www.stanford.edu/group/journal/2005/pdfs/Carl.pdf, although on my screen a lot of characters are missing - maybe they print correctly but just don't show up on my screen.

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