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

TL;DR

This paper introduces a novel factorization of Riemann's zeta function and Dedekind zeta functions of quadratic fields into two components, providing new insights and efficient computation methods.

Contribution

It presents a new factorization of the Riemann and Dedekind zeta functions into two functions with proven functional equations and explicit series expansions, including at positive even integers.

Findings

Proves a unique functional equation for the factorized functions.

Provides rapidly converging series for the functions.

Explicitly calculates series terms at positive even integers using Bernoulli numbers.

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.