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Zeta-Regularized Product

Given a positive nondecreasing sequence 0<lambda_1<=lambda_2<=..., the zeta-regularized product is defined by

product_(n=1)^^^inftylambda_n=exp(-zeta_lambda^'(0)),

where zeta_lambda(s) is the zeta function

zeta_lambda(s)=sum_(n=1)^inftylambda_n^(-s)

associated with the sequence {lambda_n} (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 zeta_lambda^'(0).

The notation product_(n=1)^^^infty appears for example in Mizuno (2006).

See also

Prime Products, Regularized Product, Zeta Function, Zeta-Regularized Sum

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References

Mizuno, Y. "Generalized Lerch Formulas: Examples of Zeta-Regularized Products." J. Number Th. 118, 155-171, 2006.Muñoz García, E. and Pérez Marco, R. "The Product Over All Primes is 4pi^2." 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 4pi^2." 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.

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Zeta-Regularized Product

Cite this as:

Weisstein, Eric W. "Zeta-Regularized Product." From MathWorld--A Wolfram Web Resource. https://mathworld.wolfram.com/Zeta-RegularizedProduct.html

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