Translated sums of primitive sets
The Erdős primitive set conjecture states that the sum $f(A) = \sum _{a\,\in \,A}\tfrac{1}{a\log a}$, 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) =...
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| Format: | Article |
| Language: | English |
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Académie des sciences
2022-04-01
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| Series: | Comptes Rendus. Mathématique |
| Online Access: | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.285/ |
| _version_ | 1797651456777519104 |
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| author | Lichtman, Jared Duker |
| author_facet | Lichtman, Jared Duker |
| author_sort | Lichtman, Jared Duker |
| collection | DOAJ |
| description | The Erdős primitive set conjecture states that the sum $f(A) = \sum _{a\,\in \,A}\tfrac{1}{a\log a}$, 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 $f(A,h) = \sum _{a\,\in \,A}\tfrac{1}{a(\log a+h)}$ is false starting at $h=81$, by comparison with semiprimes. In this note we prove that such falsehood occurs already at $h= 1.04\cdots $, and show this translate is best possible for semiprimes. We also obtain results for translated sums of $k$-almost primes with larger $k$. |
| first_indexed | 2024-03-11T16:16:06Z |
| format | Article |
| id | doaj.art-3f892c0a18c34c5e8da75dd6ee0784a4 |
| institution | Directory Open Access Journal |
| issn | 1778-3569 |
| language | English |
| last_indexed | 2024-03-11T16:16:06Z |
| publishDate | 2022-04-01 |
| publisher | Académie des sciences |
| record_format | Article |
| series | Comptes Rendus. Mathématique |
| spelling | doaj.art-3f892c0a18c34c5e8da75dd6ee0784a42023-10-24T14:19:46ZengAcadémie des sciencesComptes Rendus. Mathématique1778-35692022-04-01360G440941410.5802/crmath.28510.5802/crmath.285Translated sums of primitive setsLichtman, Jared Duker0Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UKThe Erdős primitive set conjecture states that the sum $f(A) = \sum _{a\,\in \,A}\tfrac{1}{a\log a}$, 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 $f(A,h) = \sum _{a\,\in \,A}\tfrac{1}{a(\log a+h)}$ is false starting at $h=81$, by comparison with semiprimes. In this note we prove that such falsehood occurs already at $h= 1.04\cdots $, and show this translate is best possible for semiprimes. We also obtain results for translated sums of $k$-almost primes with larger $k$.https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.285/ |
| spellingShingle | Lichtman, Jared Duker Translated sums of primitive sets Comptes Rendus. Mathématique |
| title | Translated sums of primitive sets |
| title_full | Translated sums of primitive sets |
| title_fullStr | Translated sums of primitive sets |
| title_full_unstemmed | Translated sums of primitive sets |
| title_short | Translated sums of primitive sets |
| title_sort | translated sums of primitive sets |
| url | https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.5802/crmath.285/ |
| work_keys_str_mv | AT lichtmanjaredduker translatedsumsofprimitivesets |