Probability, information theory, and prime number theory

Abstract

For any probability distribution D = { α( n)} on Z +, we define β(m) = ∑ j=1 ∞ α(jm) , the probability in D that a ‘random’ integer is a multiple of m; and γ(k) = ∑ d|k μ(d)β(d) , the probability in D that a ‘random’ integer is relatively prime to k. We specialize this general situation to three important families of distributions: D s = { α s ( n)} = { n − s | ζ( s)} for s > 1 (the Dirichlet family); L z = { α z ( n)} = {(1 − z) z n−1 } for 0 < z < 1 (the Lambert family); and U N = {α N(n)} = { 1 N , 1⩽n⩽N; 0, n > N} for N ∈ Z + (the finite uniform family). Several basic results and concepts from analytic prime number theory are revisited from the perspective of these families of probability distributions, and the Shannon entropy foreach of these families is determined.