This paper treats the questions of convergence and summability on a convex polygon $Q$ of the Dirichlet series of a function $f(z)$ which is analytic in $Q$ and continuous on $\overline Q$ . Necessary and sufficient conditions for convergence are given for the case of a square; in the general case, if the necessary conditions for convergence are satisfied, it is sufficient that the integral $\int_0^1\frac{\omega(f;t)}t\,dt$ converge. Bibliography: 7 titles.