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Upper bounds on residues of Dedekind zeta functions of non-normal totally real cubic fields

Volume 198 / 2021

Stéphane R. Louboutin Acta Arithmetica 198 (2021), 233-256 MSC: Primary 11R42; Secondary 11M20, 11R11, 11R16. DOI: 10.4064/aa200406-18-9 Published online: 19 January 2021

Abstract

Various bounds on the absolute values of $L$-functions of number fields at $s=1$ and on residues at $s=1$ of Dedekind zeta functions of a number field $\mathbb {L}$ are known. Also, better bounds depending on the splitting behavior of given prime ideals of $\mathbb {L}$ of small norms are known. These bounds involve a term $\nu _{\mathbb {L}}$ in the series expansion at $s=1$ of the Dedekind zeta function of $\mathbb {L}$. We explain why one should expect to have bounds on $\nu _{\mathbb {L}}$ which also depend on the splitting behavior in $\mathbb {L}$ of given prime integers. We explicitly do that for $\mathbb {L}$ a real quadratic number field. We deduce very good upper bounds on the residue at $s=1$ of the Dedekind zeta function of a non-normal totally real cubic number field $\mathbb {K}$, bounds depending on the splitting behavior of the prime $p=2$ in $\mathbb {K}$.

Authors

  • Stéphane R. LouboutinAix Marseille Univ, CNRS, Centrale Marseille, I2M
    Marseille, France
    e-mail

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