On the influence of the Galois group structure on the Chebyshev bias in number fields

Abstract

In this paper we produce unconditionally new instances of Galois number field extensions exhibiting strong discrepancies in the distribution of Frobenius elements among conjugacy classes of the Galois group. We first prove an inverse Galois theoretic statement showing a dichotomy between ``extreme Chebyshev biases'' and ``equal prime ideal counting''. We further introduce a group theoretic property that implies extreme biases. In the case of abelian extensions this leads to a complete characterization of Galois groups enabling extreme biases. In the case where the Galois group is a $p$-group, a simple criterion is deduced for the existence of extreme biases, and associated effective statements of Linnik type are obtained.