Order of a Dirichlet series with irregular distribution of the exponents in half-strips

Order of Dirichlet series with regular distribution of exponents in half-strips

“…In the present work we also consider a subclass of functions in 0 (Λ) having a finite order similar to the Ritt order in the classical case. The technique and ideas of work [7], where series of arbitrary growth were treated, turn out to be applicable also in the present case [9].…”

“…In the proven theorem ( ) < ∞ although this statement makes sense also in the case ( ) = ∞; we just need to consider the half-strips of form (∞, 0 ) coinciding with the half-strip Π 0 and then again 1 = 2 . But Theorem 1 is not reduced to the simple case = , where is the order of the function in the half-plane Π 0 calculated by formulae (0.5) via the coefficients, and is the order in the half-strip ( , 0 ), , see [9], [10].…”

“…In the general case we have ̸ = [12]. However, it was established in [9] that if the width of each half-strip = ( , ), = 1, 2, exceeds 2 ( ), where ( ) is the -density, then the Ritt orders in these hafl-strips are equal. As in work [4], a natural question arises: under which conditions each function in 0 (Λ) in different half-strips of width not less than 2 ( ) have the same order?…”

“…In paper [9] the aforementioned by result by A.F. Leontiev in [3] on coinciding Ritt orders in open strips containing was extended to the case, when the convergence domain of series (0.2) is the half-plane Π 0 = { = + : < 0}.…”

“…Leontiev in [3] on coinciding Ritt orders in open strips containing was extended to the case, when the convergence domain of series (0.2) is the half-plane Π 0 = { = + : < 0}. Assuming = 0 in (0.1), as in [9], we denote by 0 (Λ) the class of all analytic functions represented by Dirichlet series (0.2) converging only in the half-plane Π 0 . In the present work we also consider a subclass of functions in 0 (Λ) having a finite order similar to the Ritt order in the classical case.…”