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
Full-text PDF Open Access
View in AMS MathViewer
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 $\mathrm {li}(x)$.
- 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
- 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
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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)
