Sbornik. Mathematics, Tome 18 (1972) no. 4, pp. 589-602
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A. F. Leont'ev. On representing entire functions of several variables by Dirichlet series. Sbornik. Mathematics, Tome 18 (1972) no. 4, pp. 589-602. http://geodesic.mathdoc.fr/item/SM_1972_18_4_a3/
@article{SM_1972_18_4_a3,
author = {A. F. Leont'ev},
title = {On~representing entire functions of several variables by {Dirichlet} series},
journal = {Sbornik. Mathematics},
pages = {589--602},
year = {1972},
volume = {18},
number = {4},
language = {en},
url = {http://geodesic.mathdoc.fr/item/SM_1972_18_4_a3/}
}
TY - JOUR
AU - A. F. Leont'ev
TI - On representing entire functions of several variables by Dirichlet series
JO - Sbornik. Mathematics
PY - 1972
SP - 589
EP - 602
VL - 18
IS - 4
UR - http://geodesic.mathdoc.fr/item/SM_1972_18_4_a3/
LA - en
ID - SM_1972_18_4_a3
ER -
%0 Journal Article
%A A. F. Leont'ev
%T On representing entire functions of several variables by Dirichlet series
%J Sbornik. Mathematics
%D 1972
%P 589-602
%V 18
%N 4
%U http://geodesic.mathdoc.fr/item/SM_1972_18_4_a3/
%G en
%F SM_1972_18_4_a3
Let $F(z_1,z_2)$ be an entire function of two complex variables. Let us take the proximate order $$ \rho(r)=1+\frac{\psi(\ln r)}{\ln r},\quad\psi(u)\uparrow\infty,\quad\underset{x\to\infty}{\psi'(x)}\downarrow0,\quad\frac{\psi(x)}x\to0, $$ and then define positive numbers $\mu_k$ ($k\geqslant1$) so that $\mu_n^{s(\mu_n)}=n/\tau$, $0<\tau<\infty$. Let us choose an integer $m>2$ and form the numbers $\mu_ne^{2\pi ik/m}$ ($k=0,1,\dots,m-1$; $n=1,2,\dots$). Let $\lambda_k$ ($k\geqslant1$) be arranged these numbers in the order of decreasing modulus. For a proper choice of the function $\psi(x)$ and the number $\tau$, the representation $$ F(z_1,z_2)=\sum_{n,m=1}^\infty a_{n,m}e^{\lambda_nz_1+\lambda_mz_2} $$ holds in the whole space $\mathbf C^2$. Bibliography: 6 titles.
