Un théorème sur les séries de Dirichlet

Abstract

We considerk Dirichlet series ∑a _j (n)n ^−s(1≤j≤k),k≥2. We suppose that for eachj the series ∑a _j (n)n ^−s converges fors=s _j =σj+it _j , and that Max σj<1/(k−1). We prove that the (Dirichlet) product of these series converges uniformly on every bounded segment of the line ℜes = (σ₁+...+σ _k )/k+1−1/k and we estimate the rate of convergence. The number 1−1/k cannot be replaced by a smaller one.