Upper bounds on residues of Dedekind zeta functions of non-normal totally real cubic fields
Volume 198 / 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 $\mathbb {K}$.
