Arithmetic holomorphic functions of exponential type on the product of half planes (Q1903379)

Let \(O_A\) be the ring of algebraic integers and \(O_K\) be the ring of algebraic integers in the field \(K\) over \(\mathbb{Q}\) with degree \([K : Q] = d\). We put \(\delta = d\) if \(K \subset \mathbb{R}\) and \(\delta = d/2\) if \(K \not \subset \mathbb{R}\). Let \(H_L (z)\) be the supporting function of \(L \subset \mathbb{C}^m\) and \(L_j : = pr_j (L)\) is \(J\)-th projection of set \(L\) in \(C^m\). The main result: There exists a finite set \(\theta\) of \(O^m_A\) (direct product of \(O_A)\) having the following property: Suppose that \(0 \leq k' < 1\) and \(f(z)\) satisfies i) \(f(z)\) is holomorphic in \(\prod^m_{i = 1} \{z_i : \text{Re} z_i < - k' \}\); ii) For any \(\varepsilon > 0\) and \(\varepsilon' > 0\), there exists \(C_{\varepsilon, \varepsilon'} \geq 0\) such that \[ \bigl |f(z) \bigr |\leq C_{\varepsilon, \varepsilon'} \exp \bigl( H_L (z) + \varepsilon |z |\bigr),\;\text{Re} z_i \leq - k - \varepsilon'; \] where \(L\) is a closed convex set contained in \(\prod^m_{i = 1} \{w_i \in C : |\text{Im} w_i |\leq b_i < \pi\), \(\text{Re} w_i \geq a_i\}\); iii) \(f((-N)^m) \subset O_K\); iv) \(\limsup \log \overline {|f(-n) |}/ |n |\leq c\) \((c\) is a positive constant), where \(|n |: = n_1 + \cdots + n_m\), \(n : = (n_1, \dots, n_m) \in N^m\). If \(\tau (\exp (-l_i)) < - c (\delta - 1)\), \(1 \leq i \leq m\), then \(f(z) = \sum P_b (z_1, \dots, z_m) b^z\), where \(P_b \in K (\theta)\) \([z_1, \dots, z_m]\), \(b : = (b_1, \dots, b_m) \in \theta\) such that \(\log b_i \in - L_i\), \(1 \leq i \leq m\).