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Given a positive nondecreasing sequence 
, the zeta-regularized product
is defined by
 |
where 
is the zeta function
 |
associated with the sequence 
(Soulé et al. 1992, p. 97; Muñoz
Garcia and Pérez-Marco 2003, 2008). This formulation assumes that the zeta
function has an analytic continuation up
to 0 or else that there is some other known means of computing 
.
The notation 
appears for example in Mizuno (2006).
."
Preprint IHES/M/03/34. May 2003. http://inc.web.ihes.fr/prepub/PREPRINTS/M03/Resu/resu-M03-34.html.Muñoz
García, E. and Pérez Marco, R. "The Product Over All Primes is
."
Commun. Math. Phys. 277, 69-81, 2008.Soulé, C.;
Abramovich, D.; Burnois, J. F.; and Kramer, J. Lectures
on Arakelov Geometry. Cambridge, England: Cambridge University Press, 1992.Weisstein, Eric W. "Zeta-Regularized Product." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Zeta-RegularizedProduct.html