Sums of digits of multiples of integers

Let q is an element of N, q >= 2. For n is an element of 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; alpha, bfh) = Sigma(x < n <= x+y) exp (2i pi(alpha(1)s(q) (h(1)n) + ... + alpha(r)s(q) (h(r)n) + theta n)), with r is an element of N*, h is an element of N*r and theta is an element of r. The case r = 1 has already been studied by Gelfond and the case r >= 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 example we prove that if k is an element of N, k >= 2, there exists infinitely many integers n with exactly k prime factors and such that s(q) (n) equivalent to am (for (m, q - 1) = 1). We also obtain upper bounds of sums of the form Sigma(n <= x) exp (2i pi alpha s(q)(hn))f(n) where f is a multiplicative fonction of modulus less than 1.