Random Generators and Normal Numbers

Pursuant to the authors’ previous chaotic-dynamical model for random digits of fundamental constants [5], 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 ∑ i 1/(b ici) 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 αb,c = ∑ n=c,c2,c3,... 1/(nb ) is b-normal, thus generalizing the Stoneham class of normals [49]. Our approach also reproves b-normality for the Korobov class [34] βb,c,d, for which the summation index n above runs instead over powers c, c 2 , c 3 , . . . 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. 10-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. AMS 2000 subject classification numbers: 11K16, 11K06, 11J81, 11K45