# Article

Ufa Mathematical Journal
Volume 10, Number 4, pp. 50-63

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

Gaisin A.M., Gaisina G.A.

DOI:10.13108/2018-10-4-50

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.