Probability, information theory, and prime number theory

✍ Scribed by Solomon W. Golomb

• Book ID: 103060218
• Publisher: Elsevier Science
• Year: 1992
• Tongue: English
• Weight: 552 KB
• Volume: 106-107
• Category: Article
• ISSN: 0012-365X
• DOI: 10.1016/0012-365x(92)90549-u

✦ Synopsis

Golomb, S.W., Probability, information theory, and prime number theory, Discrete Mathematics 106/107 (1992) 219-229. For any probability distribution D = (a(n)} on Z+, we define p(m) = C>, cu(\_jm), the probability in D that a 'random' integer is a multiple of m; and y(k) = Cdlk v(d)p(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,v = {a;(n)) = {n-"/<(s)} for s > 1 (the Dirichlet family); L, = (a;(n)} = { (1 -z)z"-' ) for O N} for NE Z+ (the fide 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 for each of these families is determined.