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: \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.

Disclosure

“π The proof is completed. AI Disclosure. The proof of the logarithmic lower bound was developed with the assistance of OpenAI’s ChatGPT-5.5 Plus. After the authors formulated the problem and interacted with ChatGPT over several rounds, ChatGPT identified a suitable test function and supplied a complete proof, which the authors subse- quently verified. The original proof invoked the Baker–Harman–Pintz theorem on primes in short intervals [BHP01]; the authors later eliminated this”

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Classification

Drafting a complete proof for author revision

Multiplier

9

Verified

2026-08-23T05:22:44.107672Z

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