  A collector wishes to collect $m$ complete sets of $N$ distinct coupons. The
draws from the population are considered to be independent and identical
distributed with replacement, and the probability that a type-$j$ coupon is
drawn is noted as $p_{j}$. Let $T_{m}(N)$ the number of trials needed for this
problem. We present the asymptotics for the expectation (five terms plus an
error), the second rising moment (six terms plus an error), and the variance of
$T_{m}(N)$ (leading term), as well as its limit distribution as $N\rightarrow
\infty$, when \begin{equation*} p_{j}=\frac{a_{j}}{\sum_{j=2}^{N+1} a_{j}},
\,\,\,\text{where}\,\,\, a_{j}=\left(\ln j\right)^{-p}, \,\,p>0.
\end{equation*} These "log-Zipf" classes of coupon probabilities are not
covered by the existing literature and the present paper comes to fill this
gap. Therefore, we enlarge the classes for which the collector's problem is
solved (moments, variance, distribution).
