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 Chowla conjecture and the Sarnak conjecture」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

One of the basic general problems in analytic number theory is to understand as much as possible the fluctuations of the Möbius function , defined as when is the product of distinct primes, and zero otherwise. For instance, as takes values in , we have the trivial bound

is already equivalent to the prime number theorem , as observed by Landau (see e.g. this previous blog post for a proof), while the much stronger (and still open) improvement

已知结果和反例

is equivalent to the notorious Riemann hypothesis . There is a general Möbius pseudorandomness heuristic that suggests that the sign pattern behaves so randomly (or pseudorandomly) that one should expect a substantial amount of cancellation in sums that involve the sign fluctuation of the Möbius function in a nontrivial fashion, with the amount of cancellation present comparable to the amount that an analogous random sum would provide; cf. the probabilistic heuristic discusse

Conjecture 1 (Chowla’s conjecture) For any fixed integer and exponents , with at least one of the odd (so as not to completely destroy the sign cancellation), we have

证明或构造的主线

Note that as for any , we can reduce to the case when the take values in here. When only one of the are odd, this is essentially the prime number theorem in arithmetic progressions (after some elementary sieving), but with two or more of the are odd, the problem becomes completely open. For instance, the estimate

is morally very close to the conjectured asymptotic

阅读时建议盯住的点

for the von Mangoldt function , where is the twin prime constant ; this asymptotic in turn implies the twin prime conjecture. (To formally deduce estimates for von Mangoldt from estimates for Möbius, though, typically requires some better control on the error terms than , in particular gains of some power of are usually needed. See this previous blog post for more discussion.)

Remark 2 The Chowla conjecture resembles an assertion that, for chosen randomly and uniformly from to , the random variables become asymptotically independent of each other (in the probabilistic sense) as . However, this is not quite accurate, because some moments (namely those with all exponents even) have the “wrong” asymptotic value, leading to some unwanted correlation between the two variables. For instance, the events and have a strong correlation with each other, basic

值得单独记下的条目

• For constant sequences, this is essentially the prime number theorem (1) .
• For periodic sequences, this is essentially the prime number theorem in arithmetic progressions.
• For quasiperiodic sequences such as for some continuous , this follows from the work of Davenport.
• For nilsequences, this is a result of Ben Green and myself .
• For horocycle flows, this is a result of Bourgain, Sarnak, and Ziegler .
• For the Rudin-Shapiro sequence , I sketched out an argument at this MathOverflow post .
• To show that , it suffices to show that the expectation of the random variable , where is drawn uniformly at random from to , can be made arbitrary small by making large (and even larger).
• By the union bound and the zero topological entropy of , it suffices to show that for any bounded deterministic coefficients , the random variable concentrates with exponentially high probability.

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