Zeta Distribution

In probability theory and statistics, the zeta distribution is a discrete probability distribution defined on the positive integers. For a zeta-distributed random variable X with parameter s, the probability that X takes the value k is given by k raised to the power of negative s, divided by the Riemann zeta function evaluated at s, which serves as the normalizing constant. The distribution is closely related to Zipf's law: because the Riemann zeta function is itself the sum of every term k to the power of negative s over the positive integers, it appears here as exactly the normalization that turns that sum into a proper probability distribution, and the multiplicities of the distinct prime factors of a zeta-distributed variable are independent random variables.