Vol. 15, No. 4, 2021

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Chebyshev's bias in dihedral and generalized quaternion Galois groups

Alexandre Bailleul

Vol. 15 (2021), No. 4, 999–1041
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 D2n or (generalized) quaternion 2n 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.

Keywords
Chebyshev's bias, prime number races, Artin $L$-functions, central zeros, symplectic characters
Mathematical Subject Classification 2010
Primary: 11N05
Secondary: 11K38, 11M20, 11R42, 11R44, 11R45
Milestones
Received: 14 January 2020
Revised: 16 September 2020
Accepted: 14 October 2020
Published: 29 May 2021
Authors
Alexandre Bailleul
University of Bordeaux
IMB, UMR 5251
Bordeaux
France