## The Erdős conjecture for primitive sets

HTML articles powered by AMS MathViewer

- by
Jared Duker Lichtman and Carl Pomerance
**HTML**| PDF - Proc. Amer. Math. Soc. Ser. B
**6**(2019), 1-14

## 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 $\mathrm {li}(x)$.

## References

- C. Axler,
*New estimates for the $n$-th prime number*, arXiv:1706.03651v1 [math.NT], 2017. - William D. Banks and Greg Martin,
*Optimal primitive sets with restricted primes*, Integers**13**(2013), Paper No. A69, 10. MR**3118387** - J. Bayless, P. Kinlaw, and D. Klyve,
*Sums over primitive sets with a fixed number of prime factors*, Math. Comp., electronically published on March 5, 2019, DOI:10.1090/mcom/3416 (to appear in print). - A. S. Besicovitch,
*On the density of certain sequences of integers*, Math. Ann.**110**(1935), no. 1, 336–341. MR**1512943**, DOI 10.1007/BF01448032 - David A. Clark,
*An upper bound of $\sum 1/(a_i\log a_i)$ for primitive sequences*, Proc. Amer. Math. Soc.**123**(1995), no. 2, 363–365. MR**1243164**, DOI 10.1090/S0002-9939-1995-1243164-0 - H. Cohen,
*High precision computation of Hardy-Littlewood constants*, preprint, https://www.math.u-bordeaux.fr/$\sim$hecohen/ . - Harold G. Diamond and Kevin Ford,
*Generalized Euler constants*, Math. Proc. Cambridge Philos. Soc.**145**(2008), no. 1, 27–41. MR**2431637**, DOI 10.1017/S0305004108001187 - Pierre Dusart,
*Explicit estimates of some functions over primes*, Ramanujan J.**45**(2018), no. 1, 227–251. MR**3745073**, DOI 10.1007/s11139-016-9839-4 - Paul Erdös,
*Note on Sequences of Integers No One of Which is Divisible By Any Other*, J. London Math. Soc.**10**(1935), no. 2, 126–128. MR**1574239**, DOI 10.1112/jlms/s1-10.1.126 - P. Erdös,
*On the integers having exactly $K$ prime factors*, Ann. of Math. (2)**49**(1948), 53–66. MR**23279**, DOI 10.2307/1969113 - Paul Erdős and Zhen Xiang Zhang,
*Upper bound of $\sum 1/(a_i\log a_i)$ for primitive sequences*, Proc. Amer. Math. Soc.**117**(1993), no. 4, 891–895. MR**1116257**, DOI 10.1090/S0002-9939-1993-1116257-4 - Mitsuo Kobayashi,
*On the density of abundant numbers*, ProQuest LLC, Ann Arbor, MI, 2010. Thesis (Ph.D.)–Dartmouth College. MR**2996025** - Youness Lamzouri,
*A bias in Mertens’ product formula*, Int. J. Number Theory**12**(2016), no. 1, 97–109. MR**3455269**, DOI 10.1142/S1793042116500068 - Jared Duker Lichtman,
*The reciprocal sum of primitive nondeficient numbers*, J. Number Theory**191**(2018), 104–118. MR**3825463**, DOI 10.1016/j.jnt.2018.03.021 - Hugh L. Montgomery and Robert C. Vaughan,
*Multiplicative number theory. I. Classical theory*, Cambridge Studies in Advanced Mathematics, vol. 97, Cambridge University Press, Cambridge, 2007. MR**2378655** - H. Riesel and R. C. Vaughan,
*On sums of primes*, Ark. Mat.**21**(1983), no. 1, 46–74. MR**706639**, DOI 10.1007/BF02384300 - J. Barkley Rosser and Lowell Schoenfeld,
*Approximate formulas for some functions of prime numbers*, Illinois J. Math.**6**(1962), 64–94. MR**137689** - Michael Rubinstein and Peter Sarnak,
*Chebyshev’s bias*, Experiment. Math.**3**(1994), no. 3, 173–197. MR**1329368**, DOI 10.1080/10586458.1994.10504289 - Zhen Xiang Zhang,
*On a conjecture of Erdős on the sum $\sum _{p\leq n}1/(p\log p)$*, J. Number Theory**39**(1991), no. 1, 14–17. MR**1123165**, DOI 10.1016/0022-314X(91)90030-F - Zhen Xiang Zhang,
*On a problem of Erdős concerning primitive sequences*, Math. Comp.**60**(1993), no. 202, 827–834. MR**1181335**, DOI 10.1090/S0025-5718-1993-1181335-9

## 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
- 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
- © Copyright 2019 by the authors under Creative Commons Attribution-Noncommercial 3.0 License (CC BY NC 3.0)
- 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
- MathSciNet review: 3937344