Sommes des chiffres de multiples d'entiers

Cécile Dartyge (1) , Gérald Tenenbaum (1)

1 IECN - Institut Élie Cartan de Nancy

Let q in N, q >= 2. For n in N, denote by s_q(n) the sum of digits of n in the q-ary digital expansion. We give upper bounds for exponential sums like G(x,y, theta ; a,h) = sum_{x= 2 by Coquet and Solinas. For r >= 2, our results are more precises and significative for a wider range of h. Furthermore they are uniform in x and theta and explicits in h. The control of these parameters is crucial for various applications given in the paper. For exemple we prove that if k in N, k >=2, there exists infinitely many integers n with exactly k prime factors and such that s_q(n) md am (for (m,q-1)=1). We also obtain upper bounds of sums of the form sum_{n le x} exp (2i pi alpha s_q(hn))f(n) where f is a multiplicative fonction of modulus less than 1.

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

Domaines: Théorie des nombres [math.NT]

Cécile Dartyge, Gérald Tenenbaum. Sommes des chiffres de multiples d'entiers. Annales de l'Institut Fourier, 2005, 55, no. 7, pp.2423-2474. ⟨10.5802/aif.2166⟩. ⟨hal-00091831v2⟩