Landau 常数的计算机辅助下界
A computer-assisted lower bound for Landau's constant
- Centre for Mathematical Sciences, Lund University（隆德大学数学中心）
机构由 AI 辅助整理，请以论文原文为准。
AI总结：
本文通过计算机辅助方法，利用正 Toeplitz 矩矩阵和线性规划对偶性，将 Landau 常数和局部单叶 Bloch 常数的下界改进至 0.51 以上，并给出严格验证。
AI中文摘要：
我们证明了如下下界：\\[ B_\infty>0.51,\qquad L>0.51,\\]其中 $B_\infty$ 是局部单叶 Bloch 常数，$L$ 是 Landau 常数，这改进了 Chen 和 Shiba 的界 $\frac{1}{2}+2\cdot10^{-8}$。论证过程转向归一化局部单叶 Bloch 函数的对数 $g=\log f'$，此时 Bloch 条件变为对 $\operatorname{Re} g$ 的单侧线性约束；正 Toeplitz 矩矩阵随后为低阶对数系数提供了一个有限维外部松弛，而该松弛上的线性规划的任何非负对偶向量都证明了一个界。浮点线性规划仅用于发现这样的对偶向量；同样的对偶性也证明了控制 Taylor 容限的二阶导数界以及收缩覆盖的系数上限。一个独立的任意精度区间算术程序重建每个见证，并检查由 $1179$ 个盒子组成的归一化系数体的有限覆盖，最小验证半径为 $0.51003$。所有数值前提均使用 Arb 在 $80$ 位十进制精度下验证。
英文摘要：
We prove the lower bounds \[ B_\infty>0.51,\qquad L>0.51, \] where $B_\infty$ is the locally univalent Bloch constant and $L$ is Landau's constant, improving the bound $\frac{1}{2}+2\cdot10^{-8}$ of Chen and Shiba. The argument passes to the logarithm $g=\log f'$ of a normalized locally univalent Bloch function, where the Bloch condition becomes a one-sided linear constraint on $\operatorname{Re} g$; positive Toeplitz moment matrices then provide a finite-dimensional outer relaxation for the low logarithmic coefficients, and any non-negative dual vector for a linear program over that relaxation certifies a bound. Floating-point linear programming is used only to discover such dual vectors; the same duality also certifies the second-derivative bounds that control the Taylor allowances and the coefficient caps that contract the cover. An independent arbitrary-precision interval-arithmetic program reconstructs every witness and checks a finite cover of the normalized coefficient body by $1179$ boxes, the smallest certified radius being $0.51003$. All numerical premises were verified with Arb at $80$ decimal digits.
