Translated sums of primitive sets

Jared Duker Lichtman

doi:10.5802/crmath.285

Combinatoire, Théorie des nombres

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.

Résumé

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 $k$ .

Reçu le : 2021-08-30
Révisé le : 2021-10-14
Accepté le : 2021-10-19
Publié le : 2022-04-26

DOI : 10.5802/crmath.285

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

Affiliations des auteurs :

Jared Duker Lichtman ¹

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

Licence :

CC-BY 4.0

Droits d'auteur : Les auteurs conservent leurs droits

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     title = {Translated sums of primitive sets},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {409--414},
     publisher = {Acad\'emie des sciences, Paris},
     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/

Bibliographie
Cité par

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