
Order of Dirichlet series with regular distribution of exponents in halfstrips
A. M. Gaisin^{ab}, G. A. Gaisina^{a} ^{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 semistrips, 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 semistrips was previously obtained by A.M. Gaisin and N.N. Aitkuzhina.
It is shown that if the width of one of the two semistrips is less than $ 2 \pi b $, then the Ritt orders of the Dirichlet series in these semistrips are not equal.
Keywords:
$R$density of sequence, Dirichlet series, $R$order, semistrip, halfplane.
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English version:
Ufa Mathematical Journal, 2018, 10:4, 50–63 (PDF, 405 kB); https://doi.org/10.13108/201810450
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 halfstrips”, Ufimsk. Mat. Zh., 10:4 (2018), 51–63; Ufa Math. J., 10:4 (2018), 50–63
Citation in format AMSBIB
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\jour Ufimsk. Mat. Zh.
\yr 2018
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\issue 4
\pages 5163
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\jour Ufa Math. J.
\yr 2018
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\pages 5063
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