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「The distribution of primes in doubly densely divisible moduli」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

As in previous posts, we use the following asymptotic notation: is a parameter going off to infinity, and all quantities may depend on unless explicitly declared to be “fixed”. The asymptotic notation is then defined relative to this parameter. A quantity is said to be of polynomial size if one has , and bounded if . We also write for , and for .

The purpose of this post is to collect together all the various refinements to the second half of Zhang’s paper that have been obtained as part of the polymath8 project and present them as a coherent argument. In order to state the main result, we need to recall some definitions. If is a bounded subset of , let denote the square-free numbers whose prime factors lie in , and let denote the product of the primes in . Note by the Chinese remainder theorem that the set of primiti

已知结果和反例

for all coprime , since one can identify with the tuple for each .

If and is a natural number, we say that is -densely divisible if, for every , one can find a factor of in the interval . We say that is doubly -densely divisible if, for every , one can find a factor of in the interval such that is itself -densely divisible. We let denote the set of doubly -densely divisible natural numbers, and the set of -densely divisible numbers.

证明或构造的主线

Given any finitely supported sequence and any primitive residue class , we define the discrepancy

for any fixed , any bounded , and any primitive , where is the von Mangoldt function . Importantly, we do not require or to be fixed, in particular could grow polynomially in , and could grow exponentially in , but the implied constant in (1) would still need to be fixed (so it has to be uniform in and ). (In previous formulations of these estimates, the system of congruence was also required to obey a controlled multiplicity hypothesis, but we no longer need this hypothesis

阅读时建议盯住的点

This improves upon the previous constraint of (see this previous post ), although that latter statement was stronger in that it only required single dense divisibility rather than double dense divisibility. However, thanks to the efficiency of the sieving step of our argument, the upgrade of the single dense divisibility hypothesis to double dense divisibility costs almost nothing with respect to the parameter (which, using this constraint, gives a value of as verified in the

This estimate is deduced from three sub-estimates, which require a bit more notation to state. We need a fixed quantity .

值得单独记下的条目

• (i) A coefficient sequence is said to be at scale for some if it is supported on an interval of the form .
• (ii) A coefficient sequence at scale is said to obey the Siegel-Walfisz theorem if one has for any , any fixed , and any primitive residue class .
• (iii) A coefficient sequence at scale (relative to this choice of ) is said to be smooth if it takes the form for some smooth function supported on obeying the derivative bounds for all fixed (note that the implied constant in the notation
• (i) We say that holds if, whenever are quantities with and for some fixed , and are coefficient sequences at scales respectively, with obeying a Siegel-Walfisz theorem, we have
• (ii) We say that holds if the conclusion (7) of holds under the same hypotheses as before, except that (6) is replaced with for some sufficiently small fixed .
• (iii) We say that holds if, whenever are quantities with and and and are coefficient sequences at scales respectively, with smooth, we have

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