Upper bound of $\sum 1/(a_ i\log a_ i)$ for primitive sequences

Abstract:

A sequence $A = \{ {a_i}\}$ of positive integers ${a_1} < {a_2} < \cdots$ is said to be primitive if no term of $A$ divides any other. The senior author conjectures that, for any primitive sequence $A$, \[ \sum \limits _{a \leqslant n,a \in A} {\frac {1} {{a \log a}}} \leqslant \sum \limits _{p \leqslant n} {\frac {1} {{p \log p}}} \quad {\text {for}}\;n > 1,\] where $p$ is a variable prime. In our two previous papers we partially proved this conjecture. The main result of this paper is: for any primitive sequence $A$, \[ \sum \limits _{a \in A} {\frac {1} {{a \log a}} < 1.84.} \] We also give a necessary and sufficient condition for this conjecture, i.e., \[ \sum \limits _{b \in B} {\frac {1} {{b \log b}} \leqslant \sum {\frac {1} {{p \log p}}} } \] for any primitive sequence $B$.