Random Generators and Normal Numbers

Pursuant to the authors' previous chaotic-dynamical model for random digits of fundamental constants [Bailey and Crandall 01], we investigate a complementary, statistical picture in which pseudorandom number generators (PRNGs) are central. Some rigorous results are achieved: We establish b-normality for constants of the form Σ 1/(b mi c ni) for certain sequences (mi), (ni) of integers. This work unifies and extends previously known classes of explicit normals. We prove that for coprime b, c > 1 the constant α b,c = Σ n=c,c 2,c 3 , … 1/(nb n ) is b-normal, thus generalizing the Stoneham class of normals [Stoneham 73a]. Our approach also reproves b-normality for the Korobov class [Korobov 90] β b, c d , for which the summation index n above runs instead over powers cd , cd2 , cd3 , … with d > 1. Eventually we describe an uncountable class of explicit normals that succumb to the PRNG approach. Numbers of the α,β classes share with fundamental constants such as π, log 2 the property that isolated digits can be directly calculated, but for these new classes such computation tends to be surprisingly rapid. For example, we find that the googol-th (i.e., 10100 -th) binary bit of α2,3 is 0. We also present a collection of other results—such as digit-density results and irrationality proofs based on PRNG ideas—for various special numbers.