On the asymptotic behavior of entire Dirichlet series

O. B. Skaskiv ; M. N. Sheremeta

Sbornik. Mathematics, Tome 59 (1988) no. 2, pp. 379-396

Résumé

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.