Given a positive nondecreasing sequence 0<λ_1<=λ_2<=..., the zeta-regularized product is defined by (( product_(n = 1))^^)^∞ λ_n = exp(-ζ_λ^, (0)), where ζ_λ(s) is the zeta function ζ_λ(s) = sum_(n = 1)^∞ λ_n^(-s) associated with the sequence {λ_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 ζ_λ^, (0).

prime products | zeta function