On the Binomial Asymptotics of an Entire Dirichlet Series - Sheremeta, M. M. (Ukrains’kyi Matematychnyi Zhurnal, 2001)
Let M(σ) be the maximum modulus and let μ(σ) be the maximum term of an entire Dirichlet series with nonnegative exponents λ n increasing to ∞. We establish a condition for λ n under which the relations $$\ln {μ }\left( {{σ ,}F} \right) \leqslant Φ _1 \left( {σ } \right) + \left( {1 + o\left( 1 \right)} \right){τ }Φ _{2} \left( {σ } \right)\quad \left( {{σ } → + ∞ } \right)$$ and $$\ln M\left( {{σ ,}F} \right) \leqslant Φ _1 \left( {σ } \right) + \left( {1 + \left( 1 \right)} \right){τ }Φ _{2} \left( {σ } \right)\quad \left( {{σ } → + ∞ } \right)$$ are equivalent under certain conditions on the functions Φ1 and Φ2.