When is an infinite product of natural numbers regularizable?

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

!=k=1k=2π

and

#=k=1pk=4π2

where n# is a primorial, and pk 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 mk (e.g. the Fibonacci numbers or the central binomial coefficients), it is always possible to evaluate the infinite product

k=1mk

in the ζ-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 mk for a regularized product to exist?

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

Answer

Given an increasing sequence 0<λ1<λ2<λ3< one defines the regularized infinite product
n=1λn=exp(ζλ(0)),
where ζλ is the zeta function associated to the sequence (λn),
ζλ(s)=n=1λsn.
(See the paper: E.Munoz Garcia and R.Perez-Marco."The Product over all Primes is 4π2". 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
pprimesplogp=pprimeselog2p=exp(24ζ
where \zeta''(0) is the second derivative of Riemann's Zeta function in 0.

Attribution
Source : Link , Question Author : Timmy Turner , Answer Author : Nikos Bagis

Leave a Comment