QUARTERLY JOURNAL OF MATHEMATICS, v.65, no.4, pp.1179 - 1193
Abstract
For a family of G-fields K, we show an effective prime ideal theorem with probability 1. Namely, outside a density zero set, pi(x, K)=Li(x)+O(x/(log x)(2)) for xa parts per thousand yen(log d(K))(c) for some c depending on G. Using the effective prime ideal theorem, we show that the exponents e(K) of the ideal class groups of a family of G-fields increase along with d(K) with probability 1