On some asymptotic formulas in the theory of the ``factorisatio numerorum'' (Q2652861)

Let \(1 < a_1 < a_2< \cdots\) be a sequence of integers such that, for some \(\sigma\), \(\sum_{k=1}^\infty a_k^{-\rho}=1\) and \(\sum_{k=1}^\infty \log a_k < \infty\), but not all \(a_k\) are powers of \(a_1\). If \(l\) is a nonnegative integer and \(n\) is a positive integer, let \(T_l(n)\) be the coefficient of \(n\) in the Dirichlet series for \(\left(\sum_{k=1}^\infty a_k^{-\rho}\right)^l\). Write \(f(n)=\sum_{l=0}^\infty T_l(n)\). It is proved in an elementary way that \(n^{-\rho} \sum_{m=1}^n f(m)\) has a positive limit as \(n\to \infty\).