Effective prime ideal theorem and exponents of ideal class groups
For a family of G-fields K, we show an effective prime ideal theorem with probability 1.
Namely, outside a density zero set, π (x, K)= Li (x)+ O (x/(log x) 2) for x≥(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.
Namely, outside a density zero set, π (x, K)= Li (x)+ O (x/(log x) 2) for x≥(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.
Abstract
For a family of G-fields K, we show an effective prime ideal theorem with probability 1. Namely, outside a density zero set, π(x, K)=Li(x)+O(x/(log x)2) for x≥(log dK)c for some c depending on G. Using the effective prime ideal theorem, we show that the exponents eK of the ideal class groups of a family of G-fields increase along with dK with probability 1.
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