A note on the partial sum of bounded Dirichlet series

Abstract

Let ${\mathcal{H}}^{\infty }$ be the space of all Dirichlet series that admit a bounded holomorphic extension to the open right half-plane $\{s\in \mathbb{C}:\mathrm{Re}s>0\},$ and let $${\mathcal{S}}_N:{\mathcal{H}}^{\infty }\to {\mathcal{H}}^{\infty };\sum _{n=1}^{\infty }{a}_n{n}^{-s}\mapsto \sum _{n=1}^N{a}_n{n}^{-s}.$$ be the $N$-th partial sum operator. This note establishes the asymptotic lower bound $$\underset{N\to \infty }{\mathrm{liminf}}\frac{\Vert {\mathcal{S}}_N{\Vert }_{{\mathcal{H}}^{\infty }\to {\mathcal{H}}^{\infty }}}{\log N}\ge \frac{1}{2\pi }.$$ Together with the upper bound of R. Balasubramanian, B. Calado, and H. Queffélec, this shows that the growth of $\Vert {\mathcal{S}}_N{\Vert }_{{\mathcal{H}}^{\infty }\to {\mathcal{H}}^{\infty }}$ is of sharp logarithmic order.