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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 |
Canad. J. Math. 53(2001), 1194-1222
Published:2001-12-01
Printed: Dec 2001
Printed: Dec 2001
- Stéphane Louboutin
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 |