On a factorization of Riemann’s ζ function with respect to a quadratic field and its computation

Abstract

Let K be a quadratic field, and let ζ _K its Dedekind zeta function. In this paper we introduce a factorization of ζ _K into two functions, L ₁ and L ₂, defined as partial Euler products of ζ _K , which lead to a factorization of Riemann’s ζ function into two functions, p ₁ and p ₂. We prove that these functions satisfy a functional equation which has a unique solution, and we give series of very fast convergence to them. Moreover, when Δ _K  > 0 the general term of these series at even positive integers is calculated explicitly in terms of generalized Bernoulli numbers.