Some effective cases of the Brauer-Siegel Theorem

Let k be an algebraic number field of degree n~ and discriminant D~. We let Kk denote the residue of ~(s), " " the zeta function of k, at s = 1. One version of the Brauer-Siegel Theorem is that if k runs through a sequence of normal extensions of Q such that nk/log IDol+ 0, then log ~c~,,log I DJ -~0. This is proved by finding estimates for ~"k from above and below. The upper estimate is easy and effective. However, the lower estimate, which we write as rc~>cI~tLDkt -~J (~>0, c(~:l>O), I l l