An estimate of a Dirichlet series with real coefficients on a ray

Abstract

ABSTRACT. Best possible conditions are found under which the logarithm of the absolute value of an entire Dirichlet series with real coefficients rarely changing sign increases on the positive ray in the same way as in the whole plane. This result essentially strengthens a similar result due to Parlor concerning the growth of Taylor series on the positive ray.KEY WORDS: entire Diriehlet series, nonaecumulation condition, interpolation sequence, sequence of sign changes of the coefficients. Let f(z) = ~ anz n (z --z + iy)(1)be an entire transcendental function with real coemcients, M/(r) = max{ If(~)l : = r}, and let {p~ } be the sequence of sign changes of the coefficients, i.e., %~, ---ath < 0, where pk, = max{n < pk : a,, # 0}. Let k(t) denote the counting function of the sequence {pk}, i.e., k(t) = ~'-~v~<, 1. If the limit lira k(t) = A t---*~ t exists, then {pk} is called a measurable sequence and the number A is called its density. In [1], it was shown that if A = 0, then in each angle {z : largzl <__ e,e > 0} the function (1) is of the same order as in the whole plane. In [2], this assertion was substantially strengthened; namely, the following theorem was proved. =p.In [3], it was shown that for any sequence {pk} of natural numbers such thatthere exists an entire function of order p = 0.56 -l , bounded on the positive ray, for which {p~} is the sequence of sign changes of the coefficients. Since for measurable sequences {pk } we have 6 = A, it follows that the condition A = 0 under which Theorem A holds is, generally speaking, essential. Using the techniques proposed here as well as the results from [4], we could show that Theorem A is valid under the following (unimprovable) condition 6 = 0. But here we shall consider the case in which

Ряды Дирихле С Вещественными Коэффициентами, Неограниченные На Положительном Луче

“…Считаем, что α(t) = 0 при 0 t < p 1 . В [7] доказана следующая теорема (формулировка дается применительно к рядам (1)).…”

“…Статья посвящена изучению асимптотики рядов Дирихле (в частности, и ря-дов (1)) с вещественными коэффициентами на луче R + в случае, когда выпол-няется условие (7).…”