Combinatoire, Théorie des nombres
Translated sums of primitive sets
Lichtman, Jared Duker1
1 Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK
Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 409-414.

The Erdős primitive set conjecture states that the sum f(A)=∑ a∈A 1 aloga, 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)=∑ a∈A 1 a(loga+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⋯, and show this translate is best possible for semiprimes. We also obtain results for translated sums of k-almost primes with larger k.

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DOI : 10.5802/crmath.285
Classification : 11N25, 11Y60, 11A05, 11M32
Lichtman, Jared Duker 1

1 Mathematical Institute, University of Oxford, Oxford, OX2 6GG, UK 
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Lichtman, Jared Duker. Translated sums of primitive sets. Comptes Rendus. Mathématique, Tome 360 (2022) no. G4, pp. 409-414. doi : 10.5802/crmath.285. http://www.numdam.org/articles/10.5802/crmath.285/

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