  The generalized Dickman distribution ${\cal D}_\theta$ with parameter
$\theta>0$ is the unique solution to the distributional equality $W=_d W^*$,
where \begin{eqnarray} W^*=_d U^{1/\theta}(W+1) \qquad (1) \end{eqnarray} with
$W$ non-negative with probability one, $U \sim {\cal U}[0,1]$ independent of
$W$, and $=_d$ denoting equality in distribution. Members of this family appear
in number theory, stochastic geometry, perpetuities and the study of
algorithms. We obtain bounds in Wasserstein type distances between ${\cal
D}_\theta$ and \begin{eqnarray} W_n= \frac{1}{n} \sum_{i=1}^n Y_k B_k \qquad
(2) \end{eqnarray} where $B_1,\ldots,B_n, Y_1, \ldots, Y_n$ are independent
with $B_k \sim {\rm Ber}(1/k), E[Y_k]=k, {\rm Var}(Y_k)=\sigma_k^2$ and provide
an application to the minimal directed spanning tree in $\mathbb{R}^2$, and
also obtain such bounds when the Bernoulli variables in $(2)$ are replaced by
Poissons. We also give simple proofs and provide bounds with optimal rates for
the Dickman convergence of the weighted sums, arising in probabilistic number
theory, of the form \begin{eqnarray} S_n=\frac{1}{\log(p_n)} \sum_{k=1}^n X_k
\log(p_k) \end{eqnarray} where $(p_k)_{k \ge 1}$ is an enumeration of the prime
numbers in increasing order and $X_k$ is Geometric with parameter $(1-1/p_k)$,
Bernoulli with success probability $1/(1+p_k)$ or Poisson with mean
$\lambda_k$.
  In addition, we broaden the class of generalized Dickman distributions by
studying the fixed points of the transformation \begin{eqnarray*} s(W^*)=_d
U^{1/\theta}s(W+1) \end{eqnarray*} generalizing $(1)$, that allows the use of
non-identity utility functions $s(\cdot)$ in Vervaat perpetuities. We obtain
distributional bounds for recursive methods that can be used to simulate from
this family.
