On the irrationality of certain series (Q5918068)

This paper deals with the irrationality of certain series of the form \(\sum^\infty_{n=1}b_n\ell^{-a_n}\). It is proved in particular that if \(r\) is an integer with \(r\geq 2\), and \((a_n)\) a sequence of distinct positive integers such that each \(a_n\) is either \(r\)-free or squarefull, then \(\sum^\infty_{n=1}a_n\ell^{-a_n}\) is irrational: this confirms a conjecture of \textit{P. Erdős} [C. R. Acad. Sci., Paris, Sér. I 292, 765--768 (1981; Zbl 0466.10028)]. The proofs are nicely written and are based on some classical arithmetical tools as Dirichlet's theorem on arithmetic progressions.