陶哲轩博客写数学问题时，通常先把对象定义清楚，再给直觉、反例和证明轮廓。把「Polymath15, tenth thread: numerics update」改写成可阅读的中文笔记，重点是：问题在问什么、已知到哪一步、下一步最容易走偏在哪。原站广告、分享条和导航已去掉。

问题在问什么

This is the tenth “research” thread of the Polymath15 project to upper bound the de Bruijn-Newman constant , continuing this post . Discussion of the project of a non-research nature can continue for now in the existing proposal thread . Progress will be summarised at this Polymath wiki page .

Most of the progress since the last thread has been on the numerical side, in which the various techniques to numerically establish zero-free regions to the equation have been streamlined, made faster, and extended to larger heights than were previously possible. The best bound for now depends on the height to which one is willing to assume the Riemann hypothesis. Using the conservative verification up to height (slightly larger than) , which has been confirmed by independent

已知结果和反例

As progress seems to have stabilised, it may be time to transition to the writing phase of the Polymath15 project. (There are still some interesting research questions to pursue, such as numerically investigating the zeroes of for negative values of , but the writeup does not necessarily have to contain every single direction pursued in the project. If enough additional interesting findings are unearthed then one could always consider writing a second paper, for instance.

Below the fold is the detailed progress report on the numerics by Rudolph Dwars and Kalpesh Muchhal .

证明或构造的主线

The effectively bounded and normalised, Riemann-Siegel type asymptotic approximation for :

enables us to explore its complex zeros and to establish zero-free regions. By choosing a promising combination and , and then numerically and analytically showing that the right-hand side doesn’t vanish in the rectangular shaped “canopy” (or a point on the blue hyperbola), a new DBN upper bound will be established. Summarized in this visual:

阅读时建议盯住的点

To verify that in such a rectangular strip, we have adopted the so-called Barrier-approach that comprises of three stages (illustrated in a picture below):

So, new numerical computations are required to verify that both the Barrier at and the non-analytical part of the range are zero-free for a certain choice of .

值得单独记下的条目

• Complex zeros could also have horizontally flown into the ‘forbidden tunnel’ at high velocity. To numerically verify this hasn’t occurred, a Barrier needs to be introduced at and checked for any zeros having flown around, through or over it
• Verifying the range (or ) is done through testing that the lower bound of always stays higher than the upper bound of the error terms. This has to be done numerically up to a certain point , after which analytical proof takes over.
• The Barrier is required to have two nearby screens at and to ensure that no complex zeros could fly around it. Hence, it has the 3D structure: .
• For the numerical verification that the Barrier is zero-free, it is treated as a ‘pile’ of rectangles. For each rectangle the winding number is computed using the argument principle and Rouché’s theorem.
• For each rectangle, the number of mesh points required is decided using the -derivative, and the t-step is decided using the -derivative.
• Since and have smooth i.e. non-oscillatory behavior, using conservative numeric integrals with the Lemma 9.3 summands, , instead of the actual summation is feasible, and is significantly faster (the time complexity of estimation becomes ind
• The Lemma bound is used to find the number of ‘mollifiers’ required to make the bound positive at . We found that using primes was the max. number of primes still allowing an acceptable computational performance.
• The approximate Triangle bound evaluates faster and is used to establish the mollified (either 0 primes or only prime 2) end point before the analytical lower bound takes over.

阅读和落地时建议先做的 5 件事

1. 用自己的语言重写定义和结论，不看原文能不能说清对象是什么。
2. 找一个最小反例或边界情形，确认假设少一条会怎样。
3. 把证明拆成可独立检验的引理，每步只保留一个新想法。
4. 若涉及计算或形式化，先写可复现的小例子，再谈一般情形。
5. 记下尚未解决的缺口：缺估计、缺构造，还是缺正确的范畴。

和智能体、形式化工具怎么接

龙虾PRO做 OpenClaw 落地时，数学笔记最有用的部分往往是「可检验的步骤」：定义、反例、引理边界。智能体适合帮忙展开计算和检索，不适合代替你决定哪条假设能扔。

本文侧重全链路风控方法论。落地时请用自身业务单据做回放验证，不要把示例阈值直接当生产策略。 相关：风控体检 · 方案资源