On the growth of a class of Dirichlet series absolutely convergent in half-plane

  • L.V. Kulyavetc' Ivan Franko Lviv National University, 1 Universytetska str., 79000, Lviv, Ukraine
  • O.M. Mulyava National University of Food Technologies, 68 Volodymyrska str., 01601, Kyiv, Ukraine
Keywords: Dirichlet series, generalized order.
Published online: 2017-06-21

Abstract


In terms of generalized orders it is investigated a relation between the growth of a Dirichlet series $F(s)=\sum\limits_{n=1}^{\infty}a_n\exp\{s\lambda_n\}$ with the abscissa of asolute convergence $A\in (-\infty,+\infty)$ and the growth of Dirichlet series $F_j(s)=\sum\limits_{n=1}^{\infty}a_{n,j}\exp\{s\lambda_n\}$, $1\le j\le 2$, with the same abscissa of absolute convergence, if the coefficients $a_n$ are connected with the coefficients $a_{n,j}$ by correlation $$ \beta\left(\frac{\lambda_n}{\ln\,\left(|a_n|e^{A\lambda_n}\right)}\right)=(1+o(1)) \prod\limits_{j=1}^{m}\beta\left(\frac{\lambda_n} {\ln\,\left(|a_{n,j}|e^{A\lambda_n}\right)}\right)^{\omega_j},\  n\to\infty, $$ where $\omega_j>0$ $(1\le j\le m)$, $\sum\limits_{j=1}^{m}\omega_j=1$, and $\alpha$ is a positive slowly increasing function on $[x_0, +\infty)$.

Article metrics
PDF downloads: 14
Abstract views: 178
How to Cite
(1)
Kulyavetc’L., MulyavaO. On the Growth of a Class of Dirichlet Series Absolutely Convergent in Half-Plane. Carpathian Math. Publ. 2017, 9 (1), 63-71.