On the asymptotic behavior of entire Dirichlet series

For entire functions $F$ given by Dirichlet series
$$ F(s)=\sum_{n=0}^\infty a_ne^{s\lambda_n},\qquad0=\lambda_0<\lambda_1<\cdots<\lambda_n\uparrow+\infty\quad(n\to+\infty), $$
absolutely convergent in $\mathbf C$ some results are proved which give best possible, or close to best possible, conditions sufficient for the relation
$$ F(s)=(1+o(1))a_\nu e^{s\lambda_\nu}\qquad(s=\sigma+it) $$
as $\sigma\to+\infty$ outside some set, where $\nu=\nu(\sigma)$ is the central index of the Dirichlet series.
Bibliography: 4 titles.