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A natural problem in the context of the coupon collector's problem is thebehavior of the maximum of independent geometrically distributed randomvariables (with distinct parameters). This question has been addressed byBrennan et al. (British J. of Math. & CS. 8 (2015), 330-336). Here we provideexplicit asymptotic expressions for the moments of that maximum, as well as ofthe maximum of exponential random variables with corresponding parameters. Wealso deal with the probability of each of the variables being the maximal one.  The calculations lead to expressions involving Hurwitz's zeta function atcertain special points. We find here explicitly the values of the function atthese points. Also, the distribution function of the maximum we deal with isclosely related to the generating function of the partition function. Thus, ourresults (and proofs) rely on classical results pertaining to the partitionfunction.

Maximum of Exponential Random Variables, Hurwitz's Zeta Function, and the Partition Function