AI 中文总结
该数学研究针对有界Dirichlet级数空间上的第N个部分和算子，建立了其算子范数的渐近下界，结合已有上界证明该范数增长为精确对数阶，完善了相关算子范数的渐近刻画。
AI 中文摘要
设$\u27e8\fancyscript{H}^\fancyscript{\u221e}\u27e9$为所有可延拓为开右半平面$\boldsymbol{\u27e8s\u2208\fancyscript{C}: \u27e9Re\u2009s\u00a0>\u00a00\u27e9}$上有界全纯函数的Dirichlet级数构成的空间，设$$\u27e8\fancyscript{S}_N: \fancyscript{H}^\fancyscript{\u221e} \to \fancyscript{H}^\fancyscript{\u221e}; \u2211_{n=1}^\u221e a_n n^{-s} \u21a6 \u2211_{n=1}^N a_n n^{-s}.\u27e9$$为第$N$个部分和算子。本注记建立了渐近下界$$\u27e8\ud835\udefb\ud835\udc36\ud835\udc2f\ud835\udc2f_{N\to \u221e} \frac{\u2016\fancyscript{S}_N\u2016_{\fancyscript{H}^\fancyscript{\u221e} \to \fancyscript{H}^\fancyscript{\u221e}}}{\u27e9log\u2009N\u27e9} \u2265 \frac{1}{2\u03c0}.\u27e9$$结合R. Balasubramanian、B. Calado与H. Queffélec给出的上界，这表明$\u27e8\u2016\fancyscript{S}_N\u2016_{\fancyscript{H}^\fancyscript{\u221e}\to \fancyscript{H}^\fancyscript{\u221e}}\u27e9$的增长具有精确的对数阶。
英文摘要
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.
