An Estimate for a Dirichlet Series Whose Exponents Are Zeros of an Entire Function With Irregular Behavior

Гайсин, А. М.

doi:10.1070/sm1995v081n01abeh003619

Cited by 9 publications

(13 citation statements)

References 7 publications

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“…Nevertheless, if the coefficients of series (0.2) lie in a fixed angle {︀ = : | | < 2 }︀ , then | ( )| ( ) cos and under an appropriate choice of the strip 2 relation (0.3) follows easily (0.4). We observe that this condition on is not essential, for more details see [7], [8], where stronger results were obtained. The Fejér and Fabri conditions are also not essential.…”

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confidence: 70%

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

Гайсин¹,

Gaisina²

2018

Ufimskii Matematicheskii Zhurnal

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We study the Dirichlet series ( ) = ∞ ∑︀ =1 with positive and unboundedly increasing exponents . We assume that the sequence of the exponents Λ = { } has a finite density; we denote this density by . We suppose that the sequence Λ is regularly distributed. This is understood in the following sense: there exists a positive concave function in the convergence class such thatHere Λ( ) is the counting function of the sequence Λ. We show that if, in addition, the growth of the function is not very high, the orders of the function in the sense of Ritt in any closed semi-strips, the width of each of which is not less than 2 , are equal. Moreover, we do not impose additional restrictions for the nearness and concentration of the points . The corresponding result for open semi-strips was previously obtained by A.M. Gaisin and N.N. Aitkuzhina.It is shown that if the width of one of the two semi-strips is less than 2 , then the Ritt orders of the Dirichlet series in these semi-strips are not equal.

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confidence: 70%

“…The Fejér and Fabri conditions are also not essential. 1 For instance, in [7] the series ∞ ∑︀ =1 −1 was not assumed to be convergent even if has an arbitrarily high growth rate. Only the finiteness of a so-called -density ( ) was assumed, where is a convergence class and a condition of type…”

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confidence: 99%

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

Гайсин¹,

Gaisina²

2018

Ufimskii Matematicheskii Zhurnal

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“…Finally, it should be noted that the technique used here differs from the one used in [13, 5,6]. Here we use some methods and theories developed for series in exponentials as well as the techniques developed Lemma 1 was proved in [8].…”

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confidence: 96%

“…The sequence A = {An} is assumed given and satisfying conditions (6) and (7). Let {Pk} be the sequence of sign changes of the coefficients of the series (8). We set (9)…”

Section: Z-+oo Pmentioning

confidence: 99%

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

Гайсин

1996

Math Notes

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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

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“…Общая схема рассуждений и основные леммы. Асимптотиче-ская оценка величины M F (σ) для суммы F ряда Дирихле (4.1) через макси-мум |F | на вертикальном отрезке I определенной длины получена в [21]. При этом длина |I| отрезка I должна быть не меньше некоторой характеристи-ки, близкой к аналогичным характеристикам, позволяющим определять ради-ус полноты системы экспонент {e…”

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