On conditions for Dirichlet series absolutely convergent in a half-plane to belong to the class of convergence - Mulyava, O. M.; Sheremeta, M. M. (Ukrains’kyi Matematychnyi Zhurnal, 2008)
For a Dirichlet series $F(s) = \sum^{∞}_{n=0}a_n \exp \{sλ_n\}$ with the abscissa of absolute convergence $σ_a = 0$, let $M(σ) = \sup\{|F(σ+it)|:\;t \in {\mathbb R}\}$ and $μ(σ) = \max\{|a_n| \exp(σ λ_n):\;n ≥ 0\},\quad σ < 0.$ It is proved that the condition $\ln \ln n = o(\ln λ_n),\;n→∞$, is necessary and sufficient for equivalence of relations $\int^0_{-1}|σ|^{ρ-1}\ln M(σ)dσ < +∞$ and $\int^0_{-1}|σ|^{ρ-1}\ln μ(σ)dσ < +∞,\quad ρ > 0,$ for each such series.