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Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)



Primes in prime number races

Authors: Jared Duker Lichtman, Greg Martin and Carl Pomerance
Journal: Proc. Amer. Math. Soc. 147 (2019), 3743-3757
MSC (2010): Primary 11A05, 11N05; Secondary 11B83, 11M26
Published electronically: June 14, 2019
MathSciNet review: 3993767
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Abstract: Rubinstein and Sarnak have shown, conditional on the Riemann hypothesis (RH) and the linear independence hypothesis (LI) on the nonreal zeros of $\zeta (s)$, that the set of real numbers $x\ge 2$ for which $\pi (x)>\operatorname {li}(x)$ has a logarithmic density, which they computed to be about $2.6\times 10^{-7}$. A natural problem is to examine the actual primes in this race. We prove, assuming RH and LI, that the logarithmic density of the set of primes $p$ for which $\pi (p)>\operatorname {li}(p)$ relative to the prime numbers exists and is the same as the Rubinstein–Sarnak density. We also extend such results to a broad class of prime number races, including the “Mertens race” between $\prod _{p< x}(1-1/p)^{-1}$ and $e^{\gamma }\log x$ and the “Zhang race” between $\sum _{p\ge x}1/(p\log p)$ and $1/\log x$. These latter results resolve a question of the first and third authors from a previous paper, leading to further progress on a 1988 conjecture of Erdős on primitive sets.

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Additional Information

Jared Duker Lichtman
Affiliation: DPMMS, Centre for Mathematical Sciences, University of Cambridge, Wilberforce Road, Cambridge CB3 0WB, United Kingdom
MR Author ID: 1237291

Greg Martin
Affiliation: Department of Mathematics, University of British Columbia, Room 121, 1984 Mathematics Road, Vancouver, British Columbia, Canada V6T 1Z2
MR Author ID: 619056
ORCID: 0000-0002-8476-9495

Carl Pomerance
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
MR Author ID: 140915

Keywords: Prime number race, Chebyshev’s bias, Mertens product formula, logarithmic density, limiting distribution, primitive set, primitive sequence
Received by editor(s): September 10, 2018
Received by editor(s) in revised form: January 5, 2019
Published electronically: June 14, 2019
Additional Notes: The first-named author thanks the office for undergraduate research at Dartmouth College. He is also grateful for a Churchill Scholarship at the University of Cambridge.
The second-named author was supported in part by a National Sciences and Engineering Research Council of Canada Discovery Grant.
Communicated by: Amanda Folsom
Article copyright: © Copyright 2019 American Mathematical Society