短区间上的乘法函数：Matomaki-Radziwill 定理的分析图像

旧博客原文

原题：Multiplication function on short interval

The most important beakgrouth of analytic number theory is the new understanding of multiplication function on share interval, this result is established by Kaisa Matomäki & Maksym Radziwill. Two very young and intelligent superstars.

The main theorem in them article is :

Theorem(Matomaki,Radziwill)
As soon as  when , one has:

                    

for almost all  .

 

In my understanding of the result, the main strategy is:

Step 1:Parseval indetity, monotonically inequality

Parseval indetity, monotonically inequality, this is something about the norms of the quality we wish to charge. It is just trying to understanding

as a fuzzy thing by a more chargeable quality:

 

In fact we do a cutoff, the quality we really consider is just:

established the monotonically inequality:

In my understanding, This is a perspective of the quality, due to the quality is a multiplicative function integral on a domain () with additive structure, it could be looked as a lots of wave with the periodic given by primes, so we could do a orthogonal decomposition in the fractional space, try to prove the cutoff is a error term and we get such a monotonically inequality.

But at once we get the monotonically inequality, we could look it as a compactification process and this process still carry most of the information so lead to the inequality.

It seems something similar occur in the attack of the moments estimate of zeta function by the second author. And it is also could be looked as something similar to the  spectral decomposition with some basis come from multiplication unclear, i.e. primes.

 

Step 2: Involved by multiplication property, spectral decomposition 

I called it is “spectral decomposition”, but this is not very exact. Anyway, the thing I want to say is that for multiplication function , we have Euler-product formula:

Euler-product formula:
                      

But anyway, we do not use the whole power of multiplication just use it on primes, i.e. leads to following result:

This is a identity about the function , the point is it is not just use the multiplication at a point,i.e. , but take average at a area which is natural generated and compatible with multiplication, this identity carry a lot of information of the multiplicative property. Which is crucial to get a good estimate for the quality we consider about.

 

Step 3:from linear to multilinear , Cauchy schwarz

Now, we do not use one sets , but use several sets which is carefully chosen. And we do not consider [X,2X] with linear structure anymore , instead reconsider the decomposition:

On every it equipped with a bilinear structure. And is a very small set, $|U|=o(X)$ which is in fact have much better estimate.

Now we just use a Cauchy-Schwarz:

\sum_{i=1}^n\int_{I_i\times J_i}  \frac{1}{X^2}\int_{|log(X)|^{100}}^{\frac{X}{H}}|\sum_{n\leq X}\lambda(n)n^{it}|^2dt +\int_N |\sum_{n\leq X}\lambda(n)n^{it}|^2dt$

 

Step 4: major term estimate

 

step 5:minor term estimate

 

step 6: estimate the contribution of area which is not filled

 

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补充说明

以下是新整理的中文说明；上方旧博客原文保持不变。

Matomaki-Radziwill 的工作改变了我们对乘法函数短区间平均的理解。它说明，有界乘法函数在几乎所有短区间中的平均，通常接近其长区间平均。

1. 基本问题

设 $f$ 是有界乘法函数。我们关心

$$\frac1H\sum_{x

对多数 $x$ 的行为。传统解析数论更擅长长区间平均，而短区间要求理解局部波动。

2. Matomaki-Radziwill 定理

粗略地说，只要 $H\to\infty$ 不太慢，对几乎所有 $x\le X$，短区间平均可以由长区间信息控制。这一结果为 Chowla、Sarnak 和 pretentious multiplicative functions 提供了关键输入。

3. Dirichlet polynomial 视角

把乘法函数平均转化成 Dirichlet polynomial：

$$\sum_{n\le X}\frac{f(n)}{n^{1+it}}.$$

Parseval 型恒等式把短区间均方问题变成 $t$-空间上的积分估计。这是从 additive intervals 进入 multiplicative Fourier analysis 的桥。

4. Cut-off 与单调性

证明中需要去掉某些坏尺度，并建立类似单调性的控制：截断后的对象仍然保留主要信息。这个过程有点像 compactification，把原来粗糙的短区间平均换成更可估的频率对象。

5. 为什么它重要

短区间乘法函数估计让“乘法随机性”可以在局部尺度上使用。Sarnak 和 Chowla 的许多对数平均进展，都依赖这种把局部平均、Dirichlet polynomial 和 entropy decrement 结合起来的能力。