The Bohr inequality for ordinary Dirichlet series

We extend to the setting of Dirichlet series previous results of H. Bohr for Taylor series in one variable, themselves generalized by V. I. Paulsen, G. Popescu and D. Singh or extended to several variables by L. Aizenberg, R. P. Boas and D. Khavinson. We show in particular that, if f (s) = Sigma(infinity)(n=1) an n(-2) with \\f\\(infinity) := sup(Rs.0) \f (s)\ < infinity, then Sigma(infinity)(n=1) \a(n)\n(-2) <= \\f\\infinity and even slightly better, and Sigma(infinity)(n=1) \a(n)\n(-1\2) <= C\\f\\(infinity), C being an absolute constant.