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# Explicit Upper Bounds for Residues of Dedekind Zeta Functions and Values of $L$-Functions at $s=1$, and Explicit Lower Bounds for Relative Class Numbers of $\CM$-Fields

Published:2001-12-01
Printed: Dec 2001
• Stéphane Louboutin
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## Abstract

We provide the reader with a uniform approach for obtaining various useful explicit upper bounds on residues of Dedekind zeta functions of numbers fields and on absolute values of values at $s=1$ of $L$-series associated with primitive characters on ray class groups of number fields. To make it quite clear to the reader how useful such bounds are when dealing with class number problems for $\CM$-fields, we deduce an upper bound for the root discriminants of the normal $\CM$-fields with (relative) class number one.
 Keywords: Dedekind zeta functions, $L$-functions, relative class numbers, $\CM$-fields
 MSC Classifications: 11R42 - Zeta functions and $L$-functions of number fields [See also 11M41, 19F27] 11R29 - Class numbers, class groups, discriminants