A class of probability distributions on the integers

Abstract

Let p s(n) = n −s ζ(s) for n = 1,2,3,… and s > 1 be used to define a probability distribution P s on the positive integers. If ( m 1, m 2) = 1, then divisibility by m 1 is statistically independent of divisibility by m 2. Euler's product formula for the Zeta function becomes p s(1)= 1 ζ(s) = prob. 8(n has no prime factors)=π D(1−p −8). Many heuristic probability arguments, based on the fictitious uniform distribution on the positive integers, become rigorous statements in the distribution P s . The entropy of the distribution P s is shown (in two different ways) to be logζ(s)− sζ′(s) ζ(s) .