On~representing entire functions of several variables by Dirichlet series

A. F. Leont'ev

Sbornik. Mathematics, Tome 18 (1972) no. 4, pp. 589-602

Résumé

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.