Using large prime divisors to construct normal numbers
Jean–Marie De Koninck, Imre Kátai · Annales Universitatis Scientiarum Budapestinensis de Rolando Eötvös Nominatae Sectio computatorica · 2013
Given an integer q ≥ 2, a q-normal number is an irrational number ξ such that any preassigned sequence of digits occurs in the qary expansion of ξ at the expected frequency, namely 1/q .Let η(x) be a slowly increasing function such that log η(x) log x → 0 as x → ∞.Then, letting P (n) stand for the largest prime factor of n, set Q(n) to be the smallest prime divisor of n which is larger than η(n), while setting Q(n) = 1 if P (n) > η(n).Then, we show that the real number 0.Q(1)Q(2) . . . is a normal number in base 10.With various similar constructions, we create large families of normal numbers in any given base q ≥ 2. Finally, we consider exponential sums involving the Q(n) function.