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Transactions of the American Mathematical Society

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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
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Abstract: Let $ \pi (x;q,a)$ denote the number of primes up to $ x$ that are congruent to $ a$ modulo $ q$. A prime number race investigates the system of inequalities $ \pi (x;q,a_1) > \pi (x;q,a_2) > \cdots > \pi (x;q,a_r)$ for fixed modulus $ q$ and residue classes $ a_1, \ldots , a_r$. 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 $ x$, and moreover we expect the same to be true no matter how we permute the residue classes $ a_j$; if this is the case, and if the logarithmic density of the set of such $ x$ 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 $ L$-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 $ \mathbb{Q}$-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