On the geometric means of an entire function of several complex variables represented by multiple Dirichlet series (Q921194)

Consider the double Dirichlet series \[ (1.1)\quad f(s_ 1s_ 2)=\sum^{\infty}_{m,n=1}a_{m,n}\exp (s_ 1\lambda_ m+s_ 2\mu_ n)\quad (s_ j=\sigma_ j+it_ j) \] where \(a_{m,n}\in {\mathbb{C}}\), \(\lambda_ m\) and \(\mu_ n\) are real and \[ 0<\lambda_ 1<\lambda_ 2<...<\lambda_ m\to \infty,\quad 0<\mu_ 1<\mu_ 2<...<\mu_ n\to \infty. \] Consider the family \({\mathcal F}\) of all double Dirichlet series of the form (1.1) satisfying \[ \lim_{(m,n)\to \infty}(\log | a_{m,n}|)/(\lambda_ m+\mu_ n)=-\infty,\quad \lim_{m\to \infty}(\log m)/\lambda_ m=0,\quad \lim_{n\to \infty}(\log n)/\mu_ n=0. \] Theorem 1. For \(f\in {\mathcal F}\) we have \[ (2.1)\quad \lim_{\sigma_ 1\sigma_ 2\to \infty}\frac{g_ k(\alpha_ 1\sigma_ 1,\alpha_ 2\sigma_ 2)}{g_ k(\sigma_ 1,\sigma_ 2)e^{(k\sigma_ 1(1-\alpha_ 1)}\cdot e^{k\sigma_ 2(1-\alpha_ 2)}}=0 \] where \(0<\alpha_ 1,\alpha_ 2<1\) are constants and \[ g_ k(\sigma_ 1\sigma_ 2)=\exp \{(k^ 2/e^{k\sigma_ 1}e^{k\sigma_ 2})\int^{\sigma_ 1}_{0}\int^{\sigma_ 2}_{0}\log G(x_ 1,x_ 2)e^{kx_ 1}e^{kx_ 2}dx_ 1dx_ 2\} \] \[ G(x_ 1,x_ 2)=\exp \{\lim_{T_ 1,T_ 2\to \infty}(1/4T_ 1T_ 2)\int^{T_ 1}_{-T_ 1}\int^{T_ 2}_{-T_ 2}\log | f(x_ 1+it_ 1,x_ 2+it_ 2)| dt_ 1dt_ 2\}. \]