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Chebyshev’s bias in dihedral and generalized quaternion Galois groups
Algebra & Number Theory  (IF0.938),  Pub Date : 2021-05-29, DOI: 10.2140/ant.2021.15.999
Alexandre Bailleul

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.