## Upper bounds on $L(1,\chi )$ taking into account a finite set of prime ideals

### Volume 182 / 2018

#### Abstract

Let $\chi$ range over the non-trivial primitive characters associated with the abelian extensions ${\mathbb L}/{\mathbb K}$ of a given number field ${\mathbb K}$, i.e. over the non-trivial primitive characters on ray class groups of ${\mathbb K}$. Let $f_\chi$ be the norm of the finite part of the conductor of such a character. It is known that $\vert L(1,\chi)\vert\leq {1\over 2}\mathop{\rm Res }_{s=1}(\zeta_{\mathbb K}(s))\log f_\chi +O(1)$, where the implied constants are effective and depend on ${\mathbb K}$ only. We obtain better upper bounds by taking into account the behavior of $\chi$ at some given set $P$ of prime ideals of ${\mathbb K}$. This has been done before only in the case of ${\mathbb K} ={\mathbb Q}$. This paper is devoted to giving such improvements for any ${\mathbb K}$. As a non-trivial example, we give fully explicit bounds when ${\mathbb K}$ is an imaginary quadratic number field. We also give an application to bounds on residues of Dedekind zeta functions of non-normal cubic number fields.