Approximation of Dirichlet polynomials in cases of sparse exponents

Z. Sh. Karimov

Matematičeskie zametki, Tome 19 (1976) no. 5, pp. 691-698

Résumé

Let $0<\lambda_k\uparrow\infty$, $\sum_{k=1}^\infty\lambda_k^{-1}<\infty$, and let $\gamma$ be an analytic arc. For the Dirichlet polynomial $P(z)=\sum_1^na_ke^\lambda k^z$, in angle $E-\pi/2+\varphi_0<\arg[-(z-a)]<\pi/2-\varphi_0$, $0<\varphi<\pi/2$, $\operatorname{Re}\alpha<\beta=\max\limits_{t\in\gamma}\operatorname{Re}t$ we obtain the estimate $$ |P(z)|<A\max_{t\in\gamma}|P(t)|, $$ where $A$ depends only on angle $E$ $\{\lambda_k\}$. When $\gamma$ is a segment, an estimate was obtained by L. Schwartz.