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 $k$.

incorporated referee comments, corrected $h_{\infty }=.803...$, and cited earlier 1986 reference to the Erdős primitive set conjecture

References in corpus (2)

• Almost primes and the Banks-Martin conjecture
• Mertens' prime product formula, dissected

Cited by in corpus (2)

• A proof of the Erdős primitive set conjecture
• On Erdős sums of almost primes