The level of distribution of the sum-of-digits function of linear recurrence number systems

Abstract

Let G = (G j ) j≥0 be a strictly increasing linear recurrent sequence of integers withIt is well known that each positive integer ν can be uniquely represented by the so-called greedy expansionHere the digits are defined recursively in a way that 0In the present paper we study the sum-of-digits function s G (ν) = ε 0 (ν) + • • • + ε ℓ (ν) under certain natural assumptions on the sequence G. In particular, we determine its level of distribution x ϑ . To be more precise, we show that for r, s ∈ N with gcd(aHere ϑ = ϑ(G) ≥ 1 2 can be computed explicitly and we have ϑ(G) → 1 for a 1 → ∞. As an application we show that #{k ≤ x : s G (k) ≡ r (mod s), k has at most two prime factors} ≫ x/ log x provided that the coefficient a 1 is not too small. Moreover, using Bombieri's sieve an "almost prime number theorem" for s G follows from our result.Our work extends earlier results on the classical q-ary sum-of-digits function obtained by Fouvry and Mauduit.