Results and conjectures related to a conjecture of Erd\H{o}s concerning primitive sequences

A strictly increasing sequence $\mathscr{A}$ of positive integers is said to be primitive if no term of $\mathscr{A}$ divides any other. Erd\H{o}s showed that the series $\sum_{a \in \mathscr{A}} \frac{1}{a \log a}$, where $\mathscr{A}$ 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 $\mathscr{A}$ is the sequence of the prime numbers. The purpose of this paper is to study the Erd\H{o}s conjecture. In the first part of the paper, we give two significant conjectures which are equivalent to that of Erd\H{o}s and in the second one, we study the series of the form $\sum_{a \in \mathscr{A}} \frac{1}{a (\log a + x)}$, where $x$ is a fixed non-negative real number and $\mathscr{A}$ is a primitive sequence different from $\{1\}$. In particular, we prove that the analogue of Erd\H{o}s's conjecture for those series does not hold, at least for $x \geq 363$. At the end of the paper, we propose a more general conjecture than that of Erd\H{o}s, which concerns the preceding series, and we conclude by raising some open questions.