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

Abstract

A strictly increasing sequence $\mathcal{A}$ of positive integers is said to be primitive if no term of $\mathcal{A}$ divides any other. Erdős showed that the series ${\sum }_{a\in \mathcal{A}}\frac{1}{a\log a}$, where $\mathcal{A}$ is a primitive sequence different from $\{1\}$, 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 $\mathcal{A}$ 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 ${\sum }_{a\in \mathcal{A}}\frac{1}{a(\log a+x)}$, where $x$ is a fixed non-negative real number and $\mathcal{A}$ is a primitive sequence different from $\{1\}$. In particular, we prove that the analogue of Erdős's conjecture for those series does not hold, at least for $x\ge 363$. 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.

11 pages

Cited by in corpus (1)

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