On the Binomial Asymptotics of an Entire Dirichlet Series

Abstract

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

$$\ln {\mu }\left( {{\sigma ,}F} \right) \leqslant \Phi _1 \left( {\sigma } \right) + \left( {1 + o\left( 1 \right)} \right){\tau }\Phi _{2} \left( {\sigma } \right)\quad \left( {{\sigma } \to + \infty } \right)$$

and

$$\ln M\left( {{\sigma ,}F} \right) \leqslant \Phi _1 \left( {\sigma } \right) + \left( {1 + \left( 1 \right)} \right){\tau }\Phi _{2} \left( {\sigma } \right)\quad \left( {{\sigma } \to + \infty } \right)$$

are equivalent under certain conditions on the functions Φ₁ and Φ₂.