The Moment Zeta Function and Applications

Motivated by a probabilistic analysis of a simple game (itself inspired by a problem in computational learning theory) we introduce the moment zeta function of a probability distribution, and study in depth some asymptotic properties of the moment zeta function of those distributions supported in the interval [0, 1]. One example of such zeta functions is Riemann’s zeta function (which is the moment zeta function of the uniform distribution in [0, 1]. For Riemann’s zeta function we are able to show particularly sharp versions of our results. Introduction Consider the following setup: (Ω, μ) is a space with a probability measure μ, and ω1, . . . , ωn is a collection of measurable subsets of Ω, with μ(ωi) = pi. We play a game as follows: The jth step consists of picking a point xj ∈ Ω at random, so that after k steps we have the set Xk = {x1, . . . , xk}. The game is considered to be over when ∀i ≤ n, Xk ∩ ωi 6= Xk. We consider the duration of our game to be a random variable T = T (p1, . . . , pn), and wish to compute the expection E(p1, . . . , pn). of T. This cannot, in general, be done without knowing the measures pi1i2...ik = μ(ωi1 ∩ ωi2 ∩ · · · ∩ ωik), and in the sequel we will introduce the Independence Hypothesis: pi1i2...ik = pi1 × · · · × pik . Estimates without using the independence hypothesis are shown in the companion paper [Rivin2002]. We now assume further that we don’t actually know the measures p1, . . . , pn, but know that they themselves are a sample from some 1991 Mathematics Subject Classification. 60E07, 60F15, 60J20, 91E40, 26C10.