On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence

Abstract

For a Dirichlet series $F(s) = \sum^{\infty}_{n=0}a_n \exp \{s\lambda_n\}$ with the abscissa of absolute convergence $\sigma_a = 0$, let $M(\sigma) = \sup\{|F(\sigma+it)|:\;t \in {\mathbb R}\}$ and $\mu(\sigma) = \max\{|a_n| \exp(\sigma \lambda_n):\;n \geq 0\},\quad \sigma < 0.$ It is proved that the condition $\ln \ln n = o(\ln \lambda_n),\;n\rightarrow\infty$, is necessary and sufficient for equivalence of relations $\int^0_{-1}|\sigma|^{\rho-1}\ln M(\sigma)d\sigma < +\infty$ and $\int^0_{-1}|\sigma|^{\rho-1}\ln \mu(\sigma)d\sigma < +\infty,\quad \rho > 0,$ for each such series.