On the Stability of the Maximum Term of the Entire Dirichlet Series

Abstract

We establish necessary and sufficient conditions for logarithms of the maximal terms of the entire Dirichlet series $F(z) = \sum^{+\infty}_{n=0}a_n e^{z\lambda_n}$ and $A(z) = \sum^{+\infty}_{n=0}a_n b_n e^{z\lambda_n}$ to be asymptotically equivalent as ${\rm Re}\;z \rightarrow +\infty$ outside some set of finite measure.