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Proceedings of the American Mathematical Society Series B

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The Erdős conjecture for primitive sets


Authors: Jared Duker Lichtman and Carl Pomerance
Journal: Proc. Amer. Math. Soc. Ser. B 6 (2019), 1-14
MSC (2010): Primary 11B83; Secondary 11A05, 11N05
DOI: https://doi.org/10.1090/bproc/40
Published electronically: April 10, 2019
MathSciNet review: 3937344
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Abstract: A subset of the integers larger than 1 is primitive if no member divides another. Erdős proved in 1935 that the sum of $1/(a\log a)$ for $a$ running over a primitive set $A$ is universally bounded over all choices for $A$. In 1988 he asked if this universal bound is attained for the set of prime numbers. In this paper we make some progress on several fronts and show a connection to certain prime number “races” such as the race between $\pi (x)$ and $\mathrm {li}(x)$.


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Additional Information

Jared Duker Lichtman
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
MR Author ID: 1237291
Email: jdl.18@dartmouth.edu, jared.d.lichtman@gmail.com

Carl Pomerance
Affiliation: Department of Mathematics, Dartmouth College, Hanover, New Hampshire 03755
MR Author ID: 140915
Email: carl.pomerance@dartmouth.edu

Keywords: Primitive set, primitive sequence, Mertens’ product formula
Received by editor(s): June 6, 2018
Received by editor(s) in revised form: June 26, 2018, August 3, 2018, and August 14, 2018
Published electronically: April 10, 2019
Additional Notes: The first-named author is grateful for support from the office of undergraduate research at Dartmouth College.
Communicated by: Amanda Folsom
Article copyright: © Copyright 2019 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)