1. CJM 2013 (vol 65 pp. 1320)
|Orbital $L$-functions for the Space of Binary Cubic Forms|
We introduce the notion of orbital $L$-functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formulas in some detail. Aside from their intrinsic interest, the results from this paper are used to prove the existence of secondary terms in counting functions for cubic fields. This is worked out in a companion paper.
Keywords:binary cubic forms, prehomogeneous vector spaces, Shintani zeta functions, $L$-functions, cubic rings and fields
2. CJM 2001 (vol 53 pp. 1194)
|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|
|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 |
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