Ritt–Sugimura type theorems

At the end of the nineteenth century, E. Borel introduced the concept of the order of an entire function, and then a corresponding formula was obtained for calculating this quantity in terms of the coefficients of the Taylor expansion of this function. Later, J. Ritt extended this notion to entire functions represented by Dirichlet series with positive exponents. He also obtained a similar formula for this characteristic ($R$-order), which clearly depends on the coefficients and exponents of the Dirichlet series. In the works of A. M. Gaisin, this result was completely carried over to the case of a halfplane and also a bounded convex domain. In the latter case, the author deals with Dirichlet series with complex exponents, exponential series. In this article the relationship between the growth of the Dirichlet series and the expansion coefficients in terms of Ritt order ($R$-order) is studied. Cases when the series converges uniformly in the entire plane or only in a halfplane are considered separately. In both cases the necessary and sufficient conditions for the exponents are obtained, the fulfillment of which the corresponding formulas are correct, allowing to calculate this value through the series coefficients. All previously known results of this type were only of a sufficient character. In the case of a plane, we have shown accuracy of S. Tanaka's estimates for the $R$-order.