RELATIVE GROWTH OF ENTIRE DIRICHLET SERIES WITH DIFFERENT GENERALIZED ORDERS

For entire functions $F$ and $G$ defined by Dirichlet series with exponents increasing to $+\infty$ formulas are found for the finding the generalized order $\displaystyle \varrho_{\alpha,\beta}[F]_G = \varlimsup\limits_{\sigma\to=\infty} \frac{\alpha(M^{-1}_G(M_F(\sigma)))}{\beta(\sigma)}$ and the generalized lower order $\displaystyle \lambda_{\alpha,\beta}[F]_G=\varliminf\limits_{\sigma\to+\infty} \frac{\alpha(M^{-1}_G(M_F(\sigma)))}{\beta(\sigma)}$ of $F$ with respect to $G$, where $M_F(\sigma)=\sup\{|F(\sigma+it)|:\,t\in{\Bbb R}\}$ and $\alpha$ and $\beta$ are positive increasing to $+\infty$ functions.

Відносне зростання рядів Діріхле з різними абсцисами абсолютної збіжності

УДК 517.537.72 We study the growth of a Dirichlet series with zero abscissa of absolute convergence with respect to the entire Dirichlet series by using the generalized quantities of order and lower order where is the function inverse to and is a positive increasing function growing to

On a class of vector-valued entire Dirichlet series in n variables

We consider a class F of entire Dirichlet series in n variables, whose coefficients belong to a commutative Banach algebra E. With a well defined norm, F is proved to be a Banach algebra with identity. Further results on quasi-invertibility, spectrum and continuous linear functionals are proved for elements belonging to F.