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

Abstract

We study the Dirichlet series ( ) = ∞ ∑︀ =1 with positive and unboundedly increasing exponents . We assume that the sequence of the exponents Λ = { } has a finite density; we denote this density by . We suppose that the sequence Λ is regularly distributed. This is understood in the following sense: there exists a positive concave function in the convergence class such thatHere Λ( ) is the counting function of the sequence Λ. We show that if, in addition, the growth of the function is not very high, the orders of the function in the sense of Ritt in any closed semi-strips, the width of each of which is not less than 2 , are equal. Moreover, we do not impose additional restrictions for the nearness and concentration of the points . 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 , then the Ritt orders of the Dirichlet series in these semi-strips are not equal.