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Behavior of entire Dirichlet series of class D_(Φ) on curves of bounded K-slope
N. N. Aitkuzhinaa, A. M. Gaisinb, R. A. Gaisinb a Ufa University of Science and Technologies, Zaki Validi str. 32, 450076, Ufa, Russia
b Institute of Mathematics, Ufa Federal Research Center, RAS, Chernyshevsky str. 112, 450008, Ufa, Russia
Abstract:
We study an asymptotic behavior of the sum of an entire Dirichlet series F(s)=∑naneλns, 0<λn↑∞, on curves of a bounded K-slope naturally going to infinity. For entire transcendental functions of finite order having the form f(z)=∑nanzpn, pn∈N, Pólya showed that if the density of the sequence {pn} is zero, then for each curve γ going to infinity there exists an unbounded sequence {ξn}⊂γ such that, as ξn→∞, the relation holds: lnMf(|ξn|)∼ln|f(ξn)|; here Mf(r) is the maximum of the absolute value of the function f. Later these results were completely extended by I.D. Latypov to entire Dirichlet series of finite order and finite lower order according in the Ritt sense. A further generalization was obtained in works by N.N. Yusupova–Aitkuzhina to more general classes D(Φ) and D_(Φ) defined by the convex majorant Φ. In this paper we obtain necessary and sufficient conditions for the exponents λn ensuring that the logarithm of the absolute value of the sum of any Dirichlet series from the class D_(Φ) on the curve γ of a bounded K-slope is equivalent to the logarithm of the maximum term as σ=Res→+∞ over some asymptotic set, the upper density of which is one. We note that for entire Dirichlet series of an arbitrarily fast growth the corresponding result for the case of γ=R+ was obtained by A.M. Gaisin in 1998.
Keywords:
Dirichlet series, maximal term, curve of a bounded slope, asymptotic set.
Received: 31.01.2023
Citation:
N. N. Aitkuzhina, A. M. Gaisin, R. A. Gaisin, “Behavior of entire Dirichlet series of class D_(Φ) on curves of bounded K-slope”, Ufa Math. J., 15:3 (2023), 3–12
Linking options:
https://www.mathnet.ru/eng/ufa660https://doi.org/10.13108/2023-15-3-3 https://www.mathnet.ru/eng/ufa/v15/i3/p3
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Abstract page: | 71 | Russian version PDF: | 20 | English version PDF: | 19 | References: | 25 |
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