Chebyshev's bias in dihedral and generalized quaternion Galois groups

Abstract

We study the inequities in the distribution of Frobenius elements in Galois extensions of the rational numbers with Galois groups that are either dihedral $D_{2^n}$ or (generalized) quaternion ${ℍ}_{2^n}$ of two-power order. In the spirit of recent work of Fiorilli and Jouve (2020), we study, under natural hypotheses, some families of such extensions, in a horizontal aspect, where the degree is fixed, and in a vertical aspect, where the degree goes to infinity. Our main contribution uncovers in families of extensions a phenomenon, for which Ng (2000) gave numerical evidence: real zeros of Artin $L$-functions sometimes have a radical influence on the distribution of Frobenius elements.