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

Volume 182 / 2018

Stéphane R. Louboutin Acta Arithmetica 182 (2018), 249-269 MSC: Primary 11R42; Secondary 11M20, 11R29. DOI: 10.4064/aa170426-23-10 Published online: 12 January 2018


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.


  • Stéphane R. LouboutinAix Marseille Univ., CNRS, Centrale Marseille, I2M
    Marseille, France
    Postal address:
    Institut de Mathématiques de Marseille
    Aix Marseille Université
    163 Avenue de Luminy, Case 907
    13288 Marseille Cedex 9, France

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