On representing entire functions of several variables by Dirichlet series

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 $\mathbf C^2$.
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