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 | References | Similar Articles | Additional Information

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)