# When is an infinite product of natural numbers regularizable?

I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like

$$\infty !=\mathop{\hat{\prod}}_{k=1}^\infty k = \sqrt{2\pi}$$

and

$$\infty \#=\mathop{\hat{\prod}}_{k=1}^\infty p_k = 4\pi^2$$

where $n\#$ is a primorial, and $p_k$ is the $k$-th prime. (The expression for the infinite product of primes is proven here.) That got me wondering if, given a sequence of positive integers $m_k$ (e.g. the Fibonacci numbers or the central binomial coefficients), it is always possible to evaluate the infinite product

$$\mathop{\hat{\prod}}_{k=1}^\infty m_k$$

in the $\zeta$-regularized sense. It would seem that this would require studying the convergence and the possibility of analytically continuing the corresponding Dirichlet series, but I am not too well-versed at these things. If such a regularization is not always possible, what restrictions should be imposed on the $m_k$ for a regularized product to exist?

I'd love to read up on references for this subject. Thank you!

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How did you get $\prod k =\sqrt{2\pi}$? – Fabian Aug 2 '12 at 9:54
@Fabian, if $\zeta^\prime(s)=-\sum\limits_{j=1}^\infty \frac{\log\,k}{k^s}$, then consider $\exp(-\zeta^\prime(0))$... – Timmy Turner Aug 2 '12 at 9:57
a free online preprint about the second equation is at cds.cern.ch/record/630829 (2013-05-13) – Gottfried Helms May 13 '13 at 17:50
I don't know if it helps or if it is obvious, but via exponentiation the problem is exactly the same as a classic series. – geodude Aug 24 '13 at 11:31