Random generators and normal numbers

Pursuant to the authors' previous chaotic-dynamical model for random digits of fundamental constants [Bailey and Crandall 011, 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 Sigma(i) 1/(b(mi)c(ni)) for certain sequences (m(i)), (n(i)) of integers. This work unifies and extends previously known classes of explicit normals. We prove that for coprime b,c > 1 the constant alpha(b,c) = Sigma(n=c,c2,c3,...) 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] beta(b,c,d), for which the summation index n above runs instead over powers c(d) ,c(d2) , c(d3),... with d > 1. Eventually we describe an uncountable class of explicit normals that succumb to the PRNG approach. Numbers of the alpha,beta classes share with fundamental constants such as pi, 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(100)-th) binary bit of alpha(2,3) 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.