  We study weak convergence of a sequence of point processes to a
scale-invariant simple point process. For a deterministic sequence
$(z_n)_{n\in\mathbb{N}}$ of positive real numbers increasing to infinity as $n
\to \infty$ and a sequence $(X_k)_{k\in\mathbb{N}}$ of independent non-negative
integer-valued random variables, we consider the sequence of point processes
  \begin{equation*}
  \nu_n=\sum_{k=1}^\infty X_k \delta_{z_k/z_n}, \quad n\in \mathbb{N},
  \end{equation*} and prove that, under some general conditions, it converges
vaguely in distribution to a scale-invariant Poisson process $\eta_c$ on
$(0,\infty)$ with the intensity measure having the density $ct^{-1}$,
$t\in(0,\infty)$. An important motivating example from probabilistic number
theory relies on choosing $X_k \sim {\rm Geom}(1-1/p_k)$ and $z_k=\log p_k$,
$k\in \mathbb{N}$, where $(p_k)_{k \in \mathbb{N}}$ is an enumeration of the
primes in increasing order. We derive a general result on convergence of the
integrals $\int_0^1 t \nu_n(dt)$ to the integral $\int_0^1 t \eta_c(dt)$, the
latter having a generalized Dickman distribution, thus providing a new way of
proving Dickman convergence results.
  We extend our results to the multivariate setting and provide sufficient
conditions for vague convergence in distribution for a broad class of sequences
of point processes obtained by mapping the points from $(0,\infty)$ to
$\mathbb{R}^d$ via multiplication by i.i.d. random vectors. In addition, we
introduce a new class of multivariate Dickman distributions which naturally
extends the univariate setting.
