Sur un problème de Gelfond : la somme des chiffres des nombres premiers


In this article we answer a question proposed by Gelfond in 1968. We prove that the sum of digits of prime numbers written in a basis $q\geq 2$ is equidistributed in arithmetic progressions (except for some well known degenerate cases). We prove also that the sequence $(\alpha s_q(p))$ where $p$ runs through the prime numbers is equidistributed modulo $1$ if and only if $\alpha\in \Bbb{R}\setminus\Bbb{Q}$.