Exactness of estimates for $k$th order of Dirichlet series in a semi-strip

We study the Dirichlet series converging only in a half-plane such that their sequence of exponents admits an extension to a “regular” sequence. We proved the exactness of two-sided estimates for $k$-order of the sum of the Dirichlet series in a semi-strip whose width depends on the special distribution density of the exponents.