Order of Dirichlet series with regular distribution of exponents in half-strips

Abstract

We study the Dirichlet series $F(s)=\sum\limits_{n=1}^\infty a_n e^{\lambda_n s}$ with positive and unboundedly increasing exponents $\lambda_n$. We assume that the sequence of the exponents $\Lambda=\{\lambda_n\}$ has a finite density; we denote this density by $b$. We suppose that the sequence $\Lambda$ is regularly distributed. This is understood in the following sense: there exists a positive concave function $H$ in the convergence class such that $$ |\Lambda (t) - bt |\le H (t) \quad (t> 0) \ldotp $$ Here $ \Lambda (t) $ is the counting function of the sequence $ \Lambda $. We show that if, in addition, the growth of the function $H$ is not very high, the orders of the function $F$ in the sense of Ritt in any closed semi-strips, the width of each of which is not less than $ 2 \pi b $, are equal. Moreover, we do not impose additional restrictions for the nearness and concentration of the points $ \lambda_n $. The corresponding result for open semi-strips was previously obtained by A.M. Gaisin and N.N. Aitkuzhina. It is shown that if the width of one of the two semi-strips is less than $ 2 \pi b $, then the Ritt orders of the Dirichlet series in these semi-strips are not equal.