TITLE: When is an infinite product of natural numbers regularizable?
QUESTION [65 upvotes]: I only recently heard about the concept of $\zeta$-regularization, which allows the evaluation of things like
$$\infty != \prod_{k=1}^\infty k = \sqrt{2\pi}$$
and
$$\infty \# = \prod_{k=1}^\infty p_k = 4\pi^2$$
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
$$\prod_{k=1}^\infty m_k$$
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!

REPLY [4 votes]: Given an increasing sequence $0<\lambda_1<\lambda_2<\lambda_3<\ldots$ one defines the regularized infinite product
$$
\prod^{\infty}_{n=1}\lambda_n=\exp\left(-\zeta'_{\lambda}(0)\right),
$$
where $\zeta_{\lambda}$ is the zeta function associated to the sequence $(\lambda_n)$,
$$
\zeta_{\lambda}(s)=\sum^{\infty}_{n=1}\lambda_n^{-s}.
$$
(See the paper: E.Munoz Garcia and R.Perez-Marco."The Product over all Primes is $4\pi^2$". http://cds.cern.ch/record/630829/files/sis-2003-264.pdf )
In the paper( https://arxiv.org/ftp/arxiv/papers/0903/0903.4883.pdf ) I have evaluated the
$$
\prod_{p-primes}p^{\log p}=\prod_{p-primes}e^{\log^2p}=\exp\left(24\zeta''(0)+12\log^2(2\pi)\right)
$$
where $\zeta''(0)$ is the second derivative of Riemann's Zeta function in $0$.