A note on the partial sum of bounded Dirichlet series

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: \operatorname{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 $$ \liminf_{N\to \infty} \frac{\|\mathcal S_N\|_{\mathcal H^\infty \to \mathcal H^\infty}}{\log N} \geq \frac{1}{2π}. $$ Together with the upper bound of R. Balasubramanian, B. Calado, and H. Queffélec, this shows that the growth of $\|\mathcal S_N\|_{\mathcal H^\infty\to \mathcal H^\infty}$ is of sharp logarithmic order.