STNB

Number theory seminar in memory of F. Momose

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

Presenters

Xavi Ros

Abstract

Let $K$ be a quadratic field, and let $\zeta_K$ its Dedekind zeta function. In this talk 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.

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