On the derivative of an entire Dirichlet series

M. N. Sheremeta

Sbornik. Mathematics, Tome 65 (1990) no. 1, pp. 133-145

Résumé

For a sequence $\Lambda=\lambda_n$ of nonnegative numbers increasing to $+\infty$ let $S(\Lambda)$ denote the class of Dirichlet series $F(s)=\sum_{n=0}^\infty a_n\exp(s\lambda_n)$, $s=\sigma+it$, absolutely convergent in $\mathbf C$. If $F\in S(\Lambda)$, then let $M(\sigma)=\sup\{|F(\sigma+it)|:t\in\mathbf R\}$, $L(\sigma)=M'(\sigma)/M(\sigma)$ and $\lambda_{\nu(\sigma)}$ the central exponent. It is shown that for the relation $L(\sigma)\sim\lambda_{\nu(\sigma)}$ to hold as $0\leqslant\sigma\to+\infty$ outside some set of finite measure for each function $F\in S(\Lambda)$ it is necessary and sufficient that $\sum^\infty_{n=0}\frac1{n\lambda_n}\infty$. This condition can be weakened in the case when an additional restriction is placed on the decrease of the coefficients $a_n$. Bibliography: 10 titles.