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Euler developed (at least) two different approaches in order to calculate the values $\zeta(-m)$ of the zeta function $$\zeta(s) = \sum_{n\geq 1} \frac{1}{n^s}$$ at non-positive integers.

In one approach Euler obtained the following formula for the zeta function $$\zeta(s) = (1-2^{1-s})^{-1}\sum_{n=0}^\infty \frac1{2^{n+1}}\sum_{i=0}^n \binom{n}{i} (-1)^i \frac1{(i+1)^s} $$ by applying the so called Euler transformation to the series $\sum_{n=0}^\infty (-1)^n \frac1{(n+1)^s}$ (see for example the Wiki article Euler summation).

One great advantage of this formula is that you can immediately calculate the values at the negative integers because the infinite series reduces to a finite sum in this case.

Moreover this formula gives at the same time a meromorphic continuation of the zeta functions to the complex plane!

In another approach Euler introduced a formal parameter $t$ and showed that $$(1-2^{m+1})\zeta(-m) = (t \frac{d}{dt})^m(\frac{t}{t+1})|_{t=1} = (\frac d {dx})^m(\frac{e^x}{1+e^x})| _{x=0} $$ This is for example described in the very nice book of Hida "Elementary theory of L-functions and Eisenstein series". Further, this approach could be generalized to "L-functions of totally real number fields" by Shintani, Cassou-Nouges and others (see again Hida's book, e.g.).

My questions are now the following:

1) Can the first approach described above generalized to other 
Dedekind zeta functions? I completely lack an understanding of the Euler 
transformation. Is there a way to "understand" Euler's formula 
for the zeta function in a broader sense?

2) The Riemannian approach to the meromorphic continuation of the zeta function is based on looking at the Mellin transformation of Jacobi's $\theta$-function. It is well known, by the work of Hecke, that this approach generalizes to arbitrary Hecke L-functions. On the other hand, with this approach the information about the special values at non-positive integers of the corresponding Hecke L-function are more difficult to reveal (at least as far as I know). (EDIT: I'm aware of the fact, that calculating special values is extremely hard and one of the big challenges in number theory, I just try to understand certain structures underlying this more than beautiful area of mathematics...)

This makes we wondering whether there is a relation between the approaches of Euler and Riemann to the meromorphic continuation of the zeta function.

Is there a "principle" that relates Euler's analytic continuation to Riemann's ? 

(One major difference is of course that Euler's expression doesn't involve the archimedean factor at all! I also lack a real understanding on why the archimedean Euler factor appears in Riemann's approach.)

3) Is there a known relation between the two different approaches of Euler for 
calculating the values $\zeta(-m)$? Should one expect one?

Thank you very much for your time.

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1 Answer 1

In partial answer to 2), one can in fact get information about special values of $L$-functions via theta series. See, for example, Villegas and Zagier's paper "Square roots of central values of Hecke L-series" in the book Advances in Number Theory. (Disclaimer: I'm no expert,and it's certainly a lot more complex than Euler's approach to special values of $\zeta(s)$.)

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Thank you very much for this nice reference! –  user5831 Jul 15 '11 at 11:13

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