Asymptotic Behavior of the Sum of the Dirichlet Series of Prescribed Growth on Curves

We study the relationship between the growth and decrease of the sum of the entire Dirichlet series of finite order (in the sense of Ritt) on arbitrary curves going to infinity. For a class of exponents having a regular distribution (in a certain sense), we obtain a test for the logarithm of the maximal term to be equivalent to the logarithm of the absolute value of the sum of the Dirichlet series on at least one unbounded sequence of points of the curve.