Normality of some decimals generated by primes in a residue class

Abstract

Let F(x) be either a polynomial with real coefficients and with the leading coefficient rational or an entire function having logarithmic order α where 1 < α < 4/3 and taking real values at real x. Let q1, q2,... be the sequence of all the primes congruent to ℓ (mod k) with k > 1 a fixed integer. Let (F(q))r denote the digits in the r-adic expansion of [|F(q)|]. We show the decimal.(F(q1))r(F(q2))r..., is normal to the base r. © 2012 Elsevier Inc.