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Ufimsk. Mat. Zh., 2018, Volume 10, Issue 4, Pages 51–63 (Mi ufa447)  

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

A. M. Gaisinab, G. A. Gaisinaa

a Bashkir State University, Zaki Validi str. 32, 450074, Ufa, Russia
b Institute of Mathematics, Ufa Federal Research Center, RAS, Chernyshevsky str. 112, 450008, Ufa, Russia

Abstract: We study the Dirichlet series $F(s)=\sum\limits_{n=1}^\infty a_n e^{\lambda_n s}$ with positive and unboundedly increasing exponents $\lambda_n$. We assume that the sequence of the exponents $\Lambda=\{\lambda_n\}$ has a finite density; we denote this density by $b$. We suppose that the sequence $\Lambda$ is regularly distributed. This is understood in the following sense: there exists a positive concave function $H$ in the convergence class such that
$$ |\Lambda (t) - bt |\le H (t) \quad (t> 0) \ldotp $$
Here $ \Lambda (t) $ is the counting function of the sequence $ \Lambda $. We show that if, in addition, the growth of the function $H$ is not very high, the orders of the function $F$ in the sense of Ritt in any closed semi-strips, the width of each of which is not less than $ 2 \pi b $, are equal. Moreover, we do not impose additional restrictions for the nearness and concentration of the points $ \lambda_n $. 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 \pi b $, then the Ritt orders of the Dirichlet series in these semi-strips are not equal.

Keywords: $R$-density of sequence, Dirichlet series, $R$-order, semi-strip, half-plane.

Funding Agency Grant Number
Russian Foundation for Basic Research 17-41-02 0070_ p_a
The work is supported by RFBR (grant no. 17-41-02 0070_r_a).


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English version:
Ufa Mathematical Journal, 2018, 10:4, 50–63 (PDF, 405 kB); https://doi.org/10.13108/2018-10-4-50

Bibliographic databases:

UDC: 517.537.32
MSC: 30D10
Received: 27.07.2018

Citation: A. M. Gaisin, G. A. Gaisina, “Order of Dirichlet series with regular distribution of exponents in half-strips”, Ufimsk. Mat. Zh., 10:4 (2018), 51–63; Ufa Math. J., 10:4 (2018), 50–63

Citation in format AMSBIB
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\by A.~M.~Gaisin, G.~A.~Gaisina
\paper Order of Dirichlet series with regular distribution of exponents in half-strips
\jour Ufimsk. Mat. Zh.
\yr 2018
\vol 10
\issue 4
\pages 51--63
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\jour Ufa Math. J.
\yr 2018
\vol 10
\issue 4
\pages 50--63
\crossref{https://doi.org/10.13108/2018-10-4-50}
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