On the orders and types of certain entire functions
In this notationally complicated paper the author considers generalized Dirichlet series \[ f(s) = \sum^\infty_{n=1} P_n(s) e^{-\lambda_n s}, \] where \( P_n(s) = \sum_0^{m_n} a_{n,j} s^j \) is a polynomial of degree \( m_n \), which are ``near'' to other Dirichlet series \[ f_i(s) = \sum^\infty_{n=1} P_{n,i} (s) e^{-\lambda_{n,i} \cdot s}, i = 1, \ldots, p. \] The notion of ``near-ness'' means, that \( \lambda_{n, i} \sim \lambda_n \) and \( A_n \sim \prod_{i=1}^p (A_{i,n})^{\mu_i}, \) where \( A_n = \max | a_{n,j}| . \) The first theorem gives an upper estimate for the abscissa of convergence \( \sigma_c \) of the Dirichlet series \( \sum a_{n,m_n} e^{-\lambda_n \cdot s} \) by \( \sum_1^p \mu_i \sigma_c^{(i)} \) where \( \sigma_c^{(i)} \) denotes the abscissa of convergence of the series \( \sum a_{n,m_n}^{(i)} e^{-\lambda_{n,i} \cdot s}. \) Then the author obtains (under suitable assumptions on the Dirichlet series \( f_i(s) \)) results, which state that \( f(s) \) represents an entire function, and he gives upper and lower estimates for the upper and lower Ritt-order of \( f \) involving the corresponding orders of the \( f_i \) (and similar results on the upper and lower Ritt-type). The results are too complicated to be stated in more detail.
entire functions represented by Dirichlet series
inequalities for the abscissa of convergence
scientific article; zbMATH DE number 1267330