Instruct: Identify the main and secondary category of Arxiv papers based on the titles and abstracts
Query: 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.

30
The categories of this paper are: q-alg
29
The categories of this paper are: funct-an
7
The categories of this paper are: hep-th
10
The categories of this paper are: q-bio
12
The categories of this paper are: nucl-th
3
The categories of this paper are: physics
6
The categories of this paper are: astro-ph
19
The categories of this paper are: eess
32
The categories of this paper are: atom-ph
35
The categories of this paper are: mtrl-th
13
The categories of this paper are: quant-ph
20
The categories of this paper are: acc-phys
11
The categories of this paper are: hep-lat
26
The categories of this paper are: bayes-an
4
The categories of this paper are: cond-mat
18
The categories of this paper are: econ