Translated sums of primitive sets
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 $h= 1.04\cdots$, and show this translate is best possible for semiprimes. We also obtain results for translated sums of $k$almost primes with larger $k$.
 Publication:

arXiv eprints
 Pub Date:
 October 2021
 arXiv:
 arXiv:2110.10146
 Bibcode:
 2021arXiv211010146D
 Keywords:

 Mathematics  Number Theory;
 Mathematics  Combinatorics;
 11N25;
 11Y60 (Primary);
 11A05;
 11M32 (Secondary)
 EPrint:
 6 pages