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Chebyshev’s bias for analytic L-functions

Published online by Cambridge University Press:  22 March 2019

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

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.

Type
Research Article
Copyright
© Cambridge Philosophical Society 2019

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