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「Polymath8b: Bounded intervals with many primes, after Maynard」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

where denotes the prime. In other words, is the least quantity such that there are infinitely many intervals of length that contain or more primes. Thus, for instance, the twin prime conjecture is equivalent to the assertion that , and the prime tuples conjecture would imply that is equal to the diameter of the narrowest admissible tuple of cardinality (thus we conjecturally have , , , , , and so forth; see this web page for further continuation of this sequence).

In 2004, Goldston, Pintz, and Yildirim established the bound conditional on the Elliott-Halberstam conjecture , which remains unproven. However, no unconditional finiteness of was obtained (although they famously obtained the non-trivial bound ), and even on the Elliot-Halberstam conjecture no finiteness result on the higher was obtained either (although they were able to show on this conjecture). In the recent breakthrough of Zhang , the unconditional bound was obtained, by

已知结果和反例

With the very recent preprint of James Maynard , we have the following further substantial improvements:

Theorem 1 (Maynard’s theorem) Unconditionally, we have the following bounds:

证明或构造的主线

If one assumes the Elliott-Halberstam conjecture, we have the following improved bounds:

The final conclusion on Elliott-Halberstam is not explicitly stated in Maynard’s paper, but follows easily from his methods, as I will describe below the fold. (At around the same time as Maynard’s work, I had also begun a similar set of calculations concerning , but was only able to obtain the slightly weaker bound unconditionally.) In the converse direction, the prime tuples conjecture implies that should be comparable to . Granville has also obtained the slightly weaker ex

阅读时建议盯住的点

The arguments of Maynard avoid using the difficult partial results on (weakened forms of) the Elliott-Halberstam conjecture that were established by Zhang and then refined by Polymath8; instead, the main input is the classical Bombieri-Vinogradov theorem , combined with a sieve that is closer in spirit to an older sieve of Goldston and Yildirim , than to the sieve used later by Goldston, Pintz, and Yildirim on which almost all subsequent work is based.

The aim of the Polymath8b project is to obtain improved bounds on , and higher values of , either conditional on the Elliott-Halberstam conjecture or unconditional. The likeliest routes for doing this are by optimising Maynard’s arguments and/or combining them with some of the results from the Polymath8a project. This post is intended to be the first research thread for that purpose. To start the ball rolling, I am going to give below a presentation of Maynard’s results, with

值得单独记下的条目

• for an absolute constant and any .
• for an absolute constant and any .
• whenever is sufficiently large and .
• whenever is sufficiently large and .
• If is sufficiently large, then

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