Translated sums of primitive sets

Jared Duker Lichtman

doi:10.5802/crmath.285

Combinatorics, Number theory

Translated sums of primitive sets

Jared Duker Lichtman ¹

¹ Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

Comptes Rendus. Mathématique, Volume 360 (2022), pp. 409-414.

Abstract

The Erdős primitive set conjecture states that the sum $f (A) = \sum_{a \in A} \frac{1}{a log a}$ , ranging over any primitive set $A$ of positive integers, is maximized by the set of prime numbers. Recently Laib, Derbal, and Mechik proved that the translated Erdős conjecture for the sum $f (A, h) = \sum_{a \in A} \frac{1}{a (log a + h)}$ is false starting at $h = 81$ , by comparison with semiprimes. In this note we prove that such falsehood occurs already at $h = 1.04 \dots$ , and show this translate is best possible for semiprimes. We also obtain results for translated sums of $k$ -almost primes with larger $k$ .

Received: 2021-08-30
Revised: 2021-10-14
Accepted: 2021-10-19
Published online: 2022-04-26

DOI: 10.5802/crmath.285

Classification: 11N25, 11Y60, 11A05, 11M32

Author's affiliations:

Jared Duker Lichtman ¹

¹ Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK

License:

CC-BY 4.0

Copyrights: The authors retain unrestricted copyrights and publishing rights

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     author = {Jared Duker Lichtman},
     title = {Translated sums of primitive sets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {409--414},
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     volume = {360},
     year = {2022},
     doi = {10.5802/crmath.285},
     language = {en},
}

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Jared Duker Lichtman. Translated sums of primitive sets. Comptes Rendus. Mathématique, Volume 360 (2022), pp. 409-414. doi : 10.5802/crmath.285. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.285/

References
Cited by

[1] William D. Banks; Gref Martin Optimal primitive sets with restricted primes, Integers, Volume 13 (2013), A69, 10 pages | MR | Zbl

[2] Henri Cohen High precision computation of Hardy-Littlewood constants (https://www.math.u-bordeaux.fr/~hecohen/)

[3] Paul Erdős Note on sequences of integers no one of which is divisible by any other, J. Lond. Math. Soc., Volume 10 (1935), pp. 126-128 | DOI | MR | Zbl

[4] Ilias Laib Note on translated sum on primitive sequences, Notes Number Theory Discrete Math., Volume 27 (2021) no. 3, pp. 39-43 | DOI

[5] Ilias Laib; Abdellah Derbal; Rachid Mechik Somme translatée sur des suites primitives et la conjecture d’Erdős, C. R. Math. Acad. Sci. Paris, Volume 357 (2019) no. 5, pp. 413-417 | DOI | Zbl

[6] Jared D. Lichtman Almost primes and the Banks–Martin conjecture, J. Number Theory, Volume 211 (2020), pp. 513-529 | DOI | MR | Zbl

[7] Jared D. Lichtman Mertens’ prime product formula, dissected, Integers, Volume 21A (2021), A17, 15 pages | DOI | MR | Zbl

[8] Jared D. Lichtman; Carl Pomerance The Erdős conjecture for primitive sets, Proc. Am. Math. Soc., Volume 6 (2019), pp. 1-14 | DOI | Zbl

[9] Zhenxiang Zhang On a problem of Erdős concerning primitive sequences, Math. Comput., Volume 60 (1993) no. 202, pp. 827-834 | Zbl

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