Translated sums of primitive sets

TL;DR

This paper investigates the translated Erdős primitive set conjecture, showing that the conjecture fails for small translation values around 1.04, and extends the analysis to sums over k-almost primes.

Contribution

The paper proves the translated Erdős conjecture fails at a much smaller shift than previously known, specifically around 1.04, and explores sums over k-almost primes.

Findings

The translated conjecture fails at h ≈ 1.04 for semiprimes.

The failure point is optimal for semiprimes.

Results extend to larger k-almost primes.

Abstract

The Erd\H{o}s 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 Erd\H{o}s 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 $k$.