Growth order of sum of Dirichlet series: dependence on coefficients and exponents

Abstract

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.

Representation of analytic functions by exponential series in half-plane with given growth majorant

“…, 𝜇 > 0, as 0 < 𝑠 1; in this case 𝑑 = 1. Exactly this function is used as a comparison function in studying the class of analytic in Π + 0 functions in terms of the order 𝜌, see [8], [15]. In the considered here situation 𝜇 , 𝑏 = 𝑘 1+𝜇 .…”