Asymptotic values of entire Dirichlet
series with respect to its maximal term

For an entire Dirichlet series $F(s)= 1+\sum_{n=1}^{\infty} a_n\exp\{s\lambda_n\},\, s=\sigma+it$, with the maximal term $\mu (\sigma,F)$ and the central index $\nu(\sigma,F)$ we investigate behaviour of $F(\gamma(\tau))/\mu(\gamma(\tau))$ as $\tau\to+\infty$, where $\mu (s)=\mu (\sigma,F)\exp\{it\lambda_{\nu(\sigma,F)}\}$ and $\gamma(\tau)$ is a continuous curve such that $\Re \gamma(\tau)\to +\infty$ as $\tau\to+\infty$.

1. Gray A., Shah S. M. Asymptotic values of a holomorphic function with respect to its maximal term // Pacif. J. Math. – 1966. – V.18, № 1. – P.111–120.

2. Леонтьев А. Ф. Ряды экспонент. – М.: Наука, 1976, 536 с.

3. Hayman W. K. A generalisation of Stirling's formula // J. Reine Angew. Math. – 1956. – V. 196. – P.67–95.

4. Евграфов М. А. Асимптотические оценки и целые функции. – М.: Физматгиз, 1962. – 200 с.

5. Скасків О. Б. Максимум модуля і максимальний член цілого ряду Діріхле // Доп. АН УРСР, сер.А. – 1984. – № 11. – C.22–24.

6. Pólya G., Szeg o G. Aufgaben und Lehrsätze aus der Analysis, Zweiter Band, Springer-Verlag, Berlin-Göttingen-Heidelberg-New York. – 1964.

	Asymptotic values of entire Dirichlet series with respect to its maximal term
Author	S. I. Fedynyak, M. M. Sheremeta Faculty of Mechanics and Mathematics, Lviv Ivan Franko National University
Abstract	For an entire Dirichlet series $F(s)= 1+\sum_{n=1}^{\infty} a_n\exp\{s\lambda_n\},\, s=\sigma+it$, with the maximal term $\mu (\sigma,F)$ and the central index $\nu(\sigma,F)$ we investigate behaviour of $F(\gamma(\tau))/\mu(\gamma(\tau))$ as $\tau\to+\infty$, where $\mu (s)=\mu (\sigma,F)\exp\{it\lambda_{\nu(\sigma,F)}\}$ and $\gamma(\tau)$ is a continuous curve such that $\Re \gamma(\tau)\to +\infty$ as $\tau\to+\infty$.
Keywords	entire Dirichlet series, maximal term, continuous curve
DOI	doi:10.30970/ms.19.1.31-36
Reference	1. Gray A., Shah S. M. Asymptotic values of a holomorphic function with respect to its maximal term // Pacif. J. Math. – 1966. – V.18, № 1. – P.111–120. 2. Леонтьев А. Ф. Ряды экспонент. – М.: Наука, 1976, 536 с. 3. Hayman W. K. A generalisation of Stirling's formula // J. Reine Angew. Math. – 1956. – V. 196. – P.67–95. 4. Евграфов М. А. Асимптотические оценки и целые функции. – М.: Физматгиз, 1962. – 200 с. 5. Скасків О. Б. Максимум модуля і максимальний член цілого ряду Діріхле // Доп. АН УРСР, сер.А. – 1984. – № 11. – C.22–24. 6. Pólya G., Szeg o G. Aufgaben und Lehrsätze aus der Analysis, Zweiter Band, Springer-Verlag, Berlin-Göttingen-Heidelberg-New York. – 1964.
Pages	31-36
Volume	19
Issue	1
Year	2003
Journal	Matematychni Studii
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Table of content of issue	html

Asymptotic values of entire Dirichlet series with respect to its maximal term