# 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

and

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

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!

Given an increasing sequence $0<\lambda_1<\lambda_2<\lambda_3<\ldots$ one defines the regularized infinite product
where $\zeta_{\lambda}$ is the zeta function associated to the sequence $(\lambda_n)$,
(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 )
where $\zeta''(0)$ is the second derivative of Riemann's Zeta function in $0$.