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

L. V. Kulyavetc', O. M. Mulyava

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 $$ \beta\left(\frac{\lambda_n}{\ln\,\left(|a_n|e^{A\lambda_n}\right)}\right)=(1+o(1)) \prod\limits_{j=1}^{m}\beta\left(\frac{\lambda_n} {\ln\,\left(|a_{n,j}|e^{A\lambda_n}\right)}\right)^{\omega_j},\quad n\to\infty, $$ where $\omega_j>0$ $(1\le j\le m)$, $\sum\limits_{j=1}^{m}\omega_j=1$, and $\alpha$ - positive slowly increasing function on $[x_0, +\infty)$.


Keywords


Dirichlet series, generalized order.

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