Chebyshev’s bias in dihedral and generalized quaternion Galois groups

Bailleul, Alexandre

doi:10.2140/ant.2021.15.999

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Chebyshev's bias in dihedral and generalized quaternion Galois groups

Alexandre Bailleul

Vol. 15 (2021), No. 4, 999–1041

DOI: 10.2140/ant.2021.15.999

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.

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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

Vol. 15, No. 4, 2021