The sum-of-digits function of polynomial sequences
Résumé
Let q≥2 be an integer and sq(n) denote the sum of the digits in base q of the positive integer n. The goal of this work is to study a problem of Gelfond concerning the re-partition of the sequence (sq(P(n)))n∈ in arithmetic progressions when P∈[XŞ is such that P()⊂. We answer Gelfond's question and we show the uniform distribution modulo 1 of the sequence ( sq(P(n)))n∈ for ∈ , provided that q is a large enough prime number co-prime with the leading coefficient of P.