On a factorization of Riemann's $\zeta$ function with respect to a quadratic field and its computation

Abstract

Let $K$ be a quadratic field, and let $\zeta_K$ its Dedekind zeta function. In this paper we introduce a factorization of $\zeta_K$ into two functions, $L_1$ and $L_2$, defined as partial Euler products of $\zeta_K$, which lead to a factorization of Riemann's $\zeta$ 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 $\Delta_K>0$ the general term of these series at even positive integers is calculated explicitly in terms of generalized Bernoulli numbers.