Results and conjectures related to a conjecture of Erdős concerning primitive sequences

Abstract

A strictly increasing sequence, finite or infinite, [Formula: see text] of positive integers is said to be primitive if no term of [Formula: see text] divides any other. Erdős showed that the series [Formula: see text], where [Formula: see text] is a primitive sequence different from the finite sequence [Formula: see text], are all convergent and their sums are bounded above by an absolute constant. Besides, he conjectured that the upper bound of the preceding sums is reached when [Formula: see text] is the sequence of the prime numbers. The purpose of this paper is to study the Erdős conjecture. In the first part of the paper, we give two significant conjectures which are equivalent to that of Erdős and in the second one, we study the series of the form [Formula: see text], where [Formula: see text] is a fixed non-negative real number and [Formula: see text] is a primitive sequence different from the finite sequence [Formula: see text]. In particular, we prove that the analog of Erdős’s conjecture for these series does not hold, at least for [Formula: see text]. At the end of the paper, we propose a more general conjecture than that of Erdős, which concerns the preceding series, and we conclude by raising some open questions.