On the mean values of an entire function represented by Dirichlet series of several complex variables (Q1086394)

Let \(f(z_ 1,z_ 2)\) be an entire function represented by the Dirichlet series \[ f(z_ 1,z_ 2)=\sum^{\infty}_{m,n=0}a_{m,n} \exp (\lambda_ mz_ 1 + \mu_ nz_ 2) \] of the two complex variables \(z_ 1,z_ 2\) where \((a_{m,n})\) are complex numbers, \((\lambda_ m)\), \(m\geq 1\), \((\mu_ n)\), \(n\geq 1\) are real sequences of increasing numbers whose limits are infinity \((\lambda_ 0=\mu_ 0=0)\), and such that \[ \limsup_{m+n\to \infty} (\log (m+n))/(\lambda_ m+\mu_ n)= D < +\infty,\quad \limsup_{m+n\to \infty} (\log | a_{m,n}|)/(\lambda_ m+\mu_ n)= -\infty. \] The Ritt-order \(\rho\) and the lower order \(\lambda\) of \(f(z_ 1,z_ 2)\) is given as \[ \lim_{\sigma_ 1,\sigma_ 2\to \infty}\sup (\log \log M(\sigma_ 1,\sigma_ 2))/(\sigma_ 1+\sigma_ 2)=\rho \quad and\quad \lim_{\sigma_ 1,\sigma_ 2\to \infty}\inf (\log \log M(\sigma_ 1,\sigma_ 2))/(\sigma_ 1+\sigma_ 2)=\lambda,\quad respectively. \] If, for \(0<\sigma <\infty\) and \(k_ 1,k_ 2>0\) we define the functions \[ (1)\quad I_{\delta}(\sigma_ 1,\sigma_ 2,f)=\lim_{T\to \infty}(1/2\pi)^ 2\int^{T}_{-T}| f(\sigma_ 1+it_ 1,\sigma_ 2+it_ 2)|^{\delta}dt_ 1dt_ 2, \] \[ (2)\quad A_{\delta}(\sigma_ 1,\sigma_ 2,f)=\{I_{\delta}(\sigma_ 1,\sigma_ 2,f)\}^{1/\delta},\quad \sigma_ 1,\sigma_ 2>0, \] \[ (3)\quad G_{\sigma}(\sigma_ 1,\sigma_ 2,f)=\exp \{k_ 1k_ 2 \exp -(\sigma_ 1k_ 1+\sigma_ 2k_ 2)\int^{\sigma_ 1}_{0}\int^{\sigma_ 2}_{0}Log A_{\delta}(x_ 1,x_ 2)\exp (x_ 1k_ 1+x_ 2k_ 2)dx_ 1dx_ 2\}, \] then the author shows that the Ritt-order \(\rho\), the lower order \(\lambda\) of \(f(z_ 1,z_ 2)\) and the orders \(\rho_ 1,\rho_ 2\) with respect to \(z_ 1,z_ 2\) are expressible in terms of the functions (1), (2) and (3). The results are easily generalized to several variables.