1. CJM 2002 (vol 54 pp. 468)
|Mahler's Measure and the Dilogarithm (I) |
An explicit formula is derived for the logarithmic Mahler measure $m(P)$ of $P(x,y) = p(x)y - q(x)$, where $p(x)$ and $q(x)$ are cyclotomic. This is used to find many examples of such polynomials for which $m(P)$ is rationally related to the Dedekind zeta value $\zeta_F (2)$ for certain quadratic and quartic fields.
Categories:11G40, 11R06, 11Y35
2. CJM 1998 (vol 50 pp. 794)
|Upper bounds on $|L(1,\chi)|$ and applications |
We give upper bounds on the modulus of the values at $s=1$ of Artin $L$-functions of abelian extensions unramified at all the infinite places. We also explain how we can compute better upper bounds and explain how useful such computed bounds are when dealing with class number problems for $\CM$-fields. For example, we will reduce the determination of all the non-abelian normal $\CM$-fields of degree $24$ with Galois group $\SL_2(F_3)$ (the special linear group over the finite field with three elements) which have class number one to the computation of the class numbers of $23$ such $\CM$-fields.
Keywords:Dedekind zeta function, Dirichlet series, $\CM$-field, relative class number
Categories:11M20, 11R42, 11Y35, 11R29