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Chebyshev’s bias for analytic L-functions

Part of: Algebraic number theory: local and $p$-adic fields Discontinuous groups and automorphic forms Multiplicative number theory Arithmetic algebraic geometry

Published online by Cambridge University Press:  22 March 2019

LUCILE DEVIN
LUCILE DEVIN*
Affiliation:
Department of Mathematics and Statistics, University of Ottawa, 150 Louis Pasteur pvt, Ottawa, Ontario K1N 6N5, Canada. email: [email protected]
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Abstract

We discuss the generalizations of the concept of Chebyshev’s bias from two perspectives. First, we give a general framework for the study of prime number races and Chebyshev’s bias attached to general L-functions satisfying natural analytic hypotheses. This extends the cases previously considered by several authors and involving, among others, Dirichlet L-functions and Hasse–Weil L-functions of elliptic curves over Q. This also applies to new Chebyshev’s bias phenomena that were beyond the reach of the previously known cases. In addition, we weaken the required hypotheses such as GRH or linear independence properties of zeros of L-functions. In particular, we establish the existence of the logarithmic density of the set $ \{x \ge 2:\sum\nolimits_{p \le x} {\lambda _f}(p) \ge 0\}$ for coefficients (λf(p)) of general L-functions conditionally on a much weaker hypothesis than was previously known.

MSC classification

Primary: 11N45: Asymptotic results on counting functions for algebraic and topological structures 11F30: Fourier coefficients of automorphic forms 11S40: Zeta functions and $L$-functions
Secondary: 11G40: $L$-functions of varieties over global fields; Birch-Swinnerton-Dyer conjecture
Type
Research Article
Information
Mathematical Proceedings of the Cambridge Philosophical Society , Volume 169 , Issue 1 , July 2020 , pp. 103 - 140
Copyright
© Cambridge Philosophical Society 2019

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