  A natural problem in the context of the coupon collector's problem is the
behavior of the maximum of independent geometrically distributed random
variables (with distinct parameters). This question has been addressed by
Brennan et al. (British J. of Math. & CS. 8 (2015), 330-336). Here we provide
explicit asymptotic expressions for the moments of that maximum, as well as of
the maximum of exponential random variables with corresponding parameters. We
also deal with the probability of each of the variables being the maximal one.
  The calculations lead to expressions involving Hurwitz's zeta function at
certain special points. We find here explicitly the values of the function at
these points. Also, the distribution function of the maximum we deal with is
closely related to the generating function of the partition function. Thus, our
results (and proofs) rely on classical results pertaining to the partition
function.
