Congruences de sommes de chiffres de valeurs polynomiales

Cécile Dartyge (1) , Gérald Tenenbaum (1)

1 IECN - Institut Élie Cartan de Nancy

Let m, g, q be integers with q >= 2 and (m,q-1)=1. For natural integer n, denote by s_q(n) the sum of digits of n in the q-ary digital expansion. Given a polynomial f with integer coefficients, degree d >=1, and such that f(m) is a positive integer for all positive integers m, it is shown that there exists C=C(f,m,q)>0 such that for any integer g, and all large N, |{ 0 <= n <= N : s_q(f(n))=0 mod gm }| >= CN^{min(1,2/d!)}. In the special case m=q=2 and f(n)=n^2, the value C=1/20 is admissible.

Mots clés: sommes des chiffres; répartition dans les progressions arithmétiques

Domaines: Théorie des nombres [math.NT]

Cécile Dartyge, Gérald Tenenbaum. Congruences de sommes de chiffres de valeurs polynomiales. Bulletin of the London Mathematical Society, 2006, 38, pp.61-69. ⟨hal-00091790v3⟩