Inclusive prime number races

Authors:
Greg Martin and Nathan Ng

Journal:
Trans. Amer. Math. Soc. **373** (2020), 3561-3607

MSC (2010):
Primary 11N13, 11M26, 11K99, 60F05, 11J71

DOI:
https://doi.org/10.1090/tran/7996

Published electronically:
February 19, 2020

Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Let denote the number of primes up to that are congruent to modulo . A *prime number race* investigates the system of inequalities for fixed modulus and residue classes . The study of prime number races was initiated by Chebyshev and further studied by many others, including Littlewood, Shanks-Rényi, Knapowski-Turan, and Kaczorowski. We expect that this system of inequalities should have arbitrarily large solutions , and moreover we expect the same to be true no matter how we permute the residue classes ; if this is the case, and if the logarithmic density of the set of such exists and is positive, the prime number race is called *inclusive*. In breakthrough research, Rubinstein and Sarnak (1994) proved conditionally that every prime number race is inclusive; they assumed not only the generalized Riemann hypothesis but also a strong statement about the linear independence of the zeros of Dirichlet -functions. We show that the same conclusion can be reached assuming the generalized Riemann hypothesis and a substantially weaker linear independence hypothesis. In fact, we can assume that almost all of the zeros may be involved in -linear relations; and we can also conclude more strongly that the associated limiting distribution has mass everywhere. This work makes use of a number of ideas from probability, the explicit formula from number theory, and the Kronecker-Weyl equidistribution theorem.

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Additional Information

**Greg Martin**

Affiliation:
Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia V6T 1Z2, Canada

Email:
gerg@math.ubc.ca

**Nathan Ng**

Affiliation:
University of Lethbridge, Department of Mathematics and Computer Science, 4401 University Drive, Lethbridge, Alberta T1K 3M4, Canada

Email:
nathan.ng@uleth.ca

DOI:
https://doi.org/10.1090/tran/7996

Received by editor(s):
May 27, 2018

Received by editor(s) in revised form:
August 23, 2019, and September 10, 2019

Published electronically:
February 19, 2020

Additional Notes:
The research for this article was supported by NSERC Discovery Grants.

Article copyright:
© Copyright 2020
Greg Martin and Nathan Ng