We study the sharpness of the conditions under which the order of the sum of the Dirichlet series converging in some half-plane can be calculated by means of certain formula depending only on the coefficients and exponents. For unbounded functions analytic in the unit circle, a formula of such kind was obtained by a series of scientist in different years, in partucilar, by Govorov in 1959, by MacLane in 1966 and by Sheremeta in 1968. Later an analogue of this notion was also introduced for a Dirichlet series converging in some half-plane. But a corresponding formula for the growth order of the Dirichlet series was established by many authors under strict restrictions. In all previous formulae there were provided the conditions, which were only sufficient for the validity of this formula. In the present work we find conditions being not only sufficient but also necessary for the possibility to calculate the growth order for each Dirichlet series by means of this formula.
Let G be a bounded convex domain with a smooth boundary in which a given system of exponents is not complete. For a class of analytic functions in G that can be represented in G by a series of exponentials, we examine the behavior of coefficients of the series expansion in terms of the growth order near the boundary ∂G. We establish two-sided estimates for the order through characteristics depending only on the indices of the series of exponentials and the support function of the domain (these estimates are strong). As a consequence, we obtain a formula for calculating the growth order through the coefficients.
In the paper we obtain two results on the behavior of Dirichlet series on a real axis. The first of them concerns the lower bound for the sum of the Dirichlet series on the system of segments [ , + ]. Here the parameters > 0, > 0 are such that ↑ +∞, ↓ 0. The needed asymptotic estimates is established by means of a method based on some inequalities for extremal functions in the appropriate non-quasi-analytic Carleman class. This approach turns out to be more effective than the known traditional ways for obtaining similar estimates. The second result specifies essentially the known theorem by M.A. Evgrafov on existence of a bounded on R Dirichlet series. According to Macintyre, the sum of this series tends to zero on R. We prove a spectific estimate for the decay rate of the function in an Macintyre-Evgrafov type example.
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