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.