Normality of some decimals generated by primes in a residue class

摘要

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 ~? (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(q_1))_r(F(q_2))_r..., is normal to the base r.