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

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