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 Mobius function is strongly orthogonal to nilsequences」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

Ben Green and I have just uploaded to the arXiv our paper, “ The Möbius function is asymptotically orthogonal to nilsequences “, which is a sequel to our earlier paper “ The quantitative behaviour of polynomial orbits on nilmanifolds “, which I talked about in this post . In this paper, we apply our previous results on quantitative equidistribution polynomial orbits in nilmanifolds to settle the Möbius and nilsequences conjecture from our earlier paper , as part of our progra

between the Möbius function and any Lipschitz nilsequence f(n), by which we mean a sequence of the form for some orbit in a nilmanifold , and some Lipschitz function on that nilmanifold. (The implied constant can depend on the nilmanifold and on the Lipschitz constant of F, but it is important that it be independent of the generator g of the orbit or the base point x.) The case when f is constant is essentially the prime number theorem ; the case when f is periodic is essenti

已知结果和反例

There is an amusing way to interpret the conjecture (using the close relationship between nilsequences and bracket polynomials) as an assertion of the pseudorandomness of the Liouville function from a computational complexity perspective. Suppose you possess a calculator with the wonderful property of being infinite precision: it can accept arbitrarily large real numbers as input, manipulate them precisely, and also store them in memory. However, this calculator has two limit

Now suppose you play the following game with an opponent.

证明或构造的主线

For instance, using your calculator you can work out the first few digits of , provided of course that you entered the constants and in advance. You can also work out the leading digits of n by storing in advance, and computing the first few digits of .

Our theorem is equivalent to the assertion that as d goes to infinity (keeping the O(1) constants fixed), your probability of winning this game converges to 1/2; in other words, your calculator becomes asymptotically useless to you for the purposes of guessing whether n has an odd or even number of prime factors, and you may as well just guess randomly.

阅读时建议盯住的点

[I should mention a recent result in a similar spirit by Mauduit and Rivat; in this language, their result asserts that knowing the last few digits of the digit-sum of n does not increase your odds of guessing correctly.]

Now let me sketch the method of proof, which basically follows the Vinogradov (and Hardy-Littlewood) approach. First, we use a factorisation theorem in our equidistribution paper to prepare the nilsequence f(n) into what is basically one of two forms: either a periodic sequence (or something very close to a periodic sequence) or a totally equidistributed sequence. (Actually, it is possible to also be a hybrid of the two, in which the sequence foliates along dense arithmetic p

值得单独记下的条目

• The opponent specifies a large integer d.
• You get to enter in O(1) real constants of your choice into your calculator. These can be absolute constants such as and , or they can depend on d (e.g. you can enter in ).
• The opponent randomly selects an d-digit integer n, and enters n into one of the registers of your calculator.
• You are allowed to perform O(1) operations on your calculator and record what is displayed on the calculator’s viewscreen.
• After this, you have to guess whether the opponent’s number n had an odd or even number of prime factors (i.e. you guess .)
• If you guess correctly, you win $1; otherwise, you lose $1.

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