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So Euler employed the following identity $$\sin(z) = z \prod_{n=1}^{\infty} \left[1-\left(\frac{z}{n\pi}\right)^{2}\right]$$

to evaluate $\zeta(2n)$, for $n\in\mathbb{N}$

I'm curious if there's been any research on an elliptic generalization of Euler's trick, replacing $\sin(z)$ with the Jacobi elliptic function $\mathrm{sn}(z,k)$ -- my attempts at googling "elliptic zeta function" are somewhat frustrated by the existence of the Weierstrass zeta function, and zeta functions of an elliptic curve, and it is unclear to me whether the elliptic zeta functions mentioned in this article correspond to the same iceberg that I am suggesting above.

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Don't know if it's useful, but the only product formula for sn listed at the Wolfram functions site is this: functions.wolfram.com/EllipticFunctions/JacobiSN/08. –  Hans Lundmark Jun 14 '11 at 7:21
    
That factorization for sine is a special case of the Weierstrass factorization theorem (which applies to the Riemann $\zeta$ function similarly). A Jacobi elliptic function has poles so it isn't entire, but it is meromorphic so it can be written as the quotient of two holomorphic functions, and each of these can then be factored. Personally I'm more interested in the infinite product representation itself than analogizing Euler's zeta trick. –  anon Jul 30 '11 at 7:09
    
"as the quotient of two holomorphic functions" - more specifically, remember that the Jacobian elliptic functions are in fact ratios of theta functions. –  J. M. Jul 30 '11 at 7:42

1 Answer 1

Notice that, as an alternative to using the product formula for the sine function to evaluate $\zeta(2n)$, for $n\in\mathbb{N}$, one can just as easily work with the partial fraction decomposition of the cotangent function (obtained by logarithmically differentiating said product formula). Indeed, by differentiating everything one more time, we can evaluate $\zeta(2n)$, for $n\in\mathbb{N},$ by comparing the partial fraction decomposition of the reciprocal of $\sin^2 z$ with its Laurent expansion.

Now the elliptic analog of the reciprocal of $\sin^2 z$ is Weierstrass's elliptic function (Weierstrass's $\wp$-function). So one needs to compare the partial fraction decomposition and the Laurent expansion of the $\wp$-function - I recommend the MathWorld article for details on this. There it shows that the coefficients in the Laurent expansion of the $\wp$-function may be expressed as a polynomial with rational coefficients in terms of 2 quantities known as the elliptic invariants - essentially the elliptic analogs of $\zeta(4)$ and $\zeta(6)$ (see equations (24) and (25)). Thus the elliptic analog of $\zeta(2n)$ is a polynomial (with rational coefficients) in the two elliptic invariants.

This is then an elliptic analog of Euler's result that $\zeta(2n)$ is a rational multiple of the n-th power of $\zeta(2).$

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