A class of probability distributions on the integers

AbstractLet ps(n) = n−sζ(s) for n = 1,2,3,… and s > 1 be used to define a probability distribution Ps on the positive integers. If (m1, m2) = 1, then divisibility by m1 is statistically independent of divisibility by m2. Euler's product formula for the Zeta function becomes ps(1)=1ζ(s)=prob.8(n has no prime factors)=πD(1−p−8). Many heuristic probability arguments, based on the fictitious uniform distribution on the positive integers, become rigorous statements in the distribution Ps. The entropy of the distribution Ps is shown (in two different ways) to be logζ(s)−sζ′(s)ζ(s)