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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!

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    How did you get $\prod k =\sqrt{2\pi}$?2012-08-02
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    @Fabian, if $\zeta^\prime(s)=-\sum\limits_{j=1}^\infty \frac{\log\,k}{k^s}$, then consider $\exp(-\zeta^\prime(0))$...2012-08-02
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    a free online preprint about the second equation is at http://cds.cern.ch/record/630829 (2013-05-13)2013-05-13
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    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.2013-08-24
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    It seems to me that a physical interpretation of regularization would be to "measure" an object as it passes through the entrance to a black hole at a velocity asymptotically approaching the speed of light.2016-01-20
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    @TimmyTurner It very much bothers me that $(-1/2)!=\Gamma(1/2)=\sqrt{2\pi}$, and $\sum_{n=1}^{-1/2}1=-1/2$. I almost feel like some divergent series have a strange "plug in -1/2" type of nature. Think it is question worthy?2016-07-27
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    It looks like an analog of Ramanujan summation to me: $\infty!=\Pi_{k=1}^\infty k=\exp\Sigma_{k=1}^\infty\log k$. https://en.wikipedia.org/wiki/Ramanujan_summation2016-10-22
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    http://math.stackexchange.com/questions/1325169/zeta-regulated-product-solving-without-the-zeta-function?rq=1 My own shot about the subject without any background knowlegde cause i was just as curious. I know i should add some constrains for the methode.2016-11-28

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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 $0$.