Euler proved that $$\sum_{n=0}^\infty \frac{1}{n^2} = \frac{{\pi}^2}{6}$$ by comparing the $z^3$ term in the power series expression of $\sin(z)$ given by

$$\sin(z) = z - \frac{z^3}{3!} + \frac{z^5}{5!} - \frac{z^7}{7!} \cdots$$

with $z^3$ term obtained by multiplying out the Hadamard product expansion for $\sin(z)$:

$$\sin(z) = z\left(1 - \frac{z^2}{\pi^2} \right)\left(1 - \frac{z^2}{4\pi^2} \right)\left(1 - \frac{z^2}{9\pi^2} \right)\cdots $$

and more generally, showed that the numbers

$$\sum_{n=0}^\infty \frac{1}{n^{2k}}$$

are rational multiples of $\pi^{2k}$, giving an algorithm to compute the rational number, by comparing the coefficients of $z^{2j + 1}$ in the sum and the expansion of the product, for each $j \leq k$, together with Newton's identities. Euler's results can be generalized to results about special values of $L$-functions outside of the critical strip, for example, Siegel and Klingen gave such a generalization for special values of Dedekind zeta functions of totally real number fields. However, the method of Siegel and Klingen is different from Euler's.

What is the natural generalization of Euler's method of proof of the results mentioned above, and what sorts of results does one obtain from it?

The sine function is periodic and one could look at doubly periodic functions on the complex plane and try to do the same things, but such functions are not holomorphic and so don't have Hadamard product expansions. One could look at functions of several complex variables that are periodic with respect to each variable, but then it's not immediately clear what such functions look like, or whether one would get anything new.

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