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 $latex H\to \infty$ when $latex x\to \infty$, one has:
$latex \sum_{x\leq n\leq x+H}\lambda(n)= o(H)$
for almost all $latex x\sim X$ .
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 $latex L^2$ norms of the quality we wish to charge. It is just trying to understanding
$latex \frac{1}{X}\int_{X}^{2X}|\frac{1}{H}\sum_{x\leq n\leq x+H}\lambda(n)|^2dx$
as a fuzzy thing by a more chargeable quality:
$latex \frac{1}{X^2}\int_{0}^{\infty}|\sum_{n\leq X}\lambda(n)n^{it}|^2dt$
In fact we do a cutoff, the quality we really consider is just:
$latex \frac{1}{X^2}\int_{|log(X)|^{100}}^{\frac{X}{H}}|\sum_{n\leq X}\lambda(n)n^{it}|^2dt$
established the monotonically inequality:
$latex \frac{1}{X}\int_{X}^{2X}|\frac{1}{H}\sum_{x\leq n\leq x+H}\lambda(n)|^2dx << \frac{1}{X^2}\int_{|log(X)|^{100}}^{\frac{X}{H}}|\sum_{n\leq X}\lambda(n)n^{it}|^2dt$
In my understanding, This is a perspective of the quality, due to the quality is a multiplicative function integral on a domain ($latex \mathbb N^*$) 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 $latex \lambda(n)$, we have Euler-product formula:
Euler-product formula:
$latex \Pi_{p,prime}(\frac{1}{1-\frac{\lambda(p)}{p^s}})=\sum_{n=1}^{\infty} \frac{\lambda(n)}{n^s}$
But anyway, we do not use the whole power of multiplication just use it on primes, i.e. $latex \lambda(pn)=\lambda(p)\lambda(n)$ leads to following result:
$latex \lambda(n)=\sum_{n=pm,p\in I}\frac{\lambda(p)\lambda(m)}{\# \{p|n, p\in I\}+1}+\lambda(n)1_{p|n;p\notin I}$
This is a identity about the function $latex \lambda(n)$, the point is it is not just use the multiplication at a point,i.e. $latex \lambda(mn)=\lambda(m)\lambda(n)$, 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 $latex I$, but use several sets $latex I_1,…,I_n $ which is carefully chosen. And we do not consider [X,2X] with linear structure anymore , instead reconsider the decomposition:
$latex [X,2X]=\amalg_{i=1}^n (I_i\times J_i) \amalg U$
On every $latex I_i\times J_i$ it equipped with a bilinear structure. And $latex U$ is a very small set, $|U|=o(X)$ which is in fact have much better estimate.
$latex \int_{|log(X)|^{100}}^{\frac{X}{H}}|\sum_{n\leq X}\lambda(n)n^{it}|^2dt =\sum_{i=1}^n\int_{I_i\times J_i} \frac{1}{X^2}\int_{|log(X)|^{100}}^{\frac{X}{H}}|\sum_{n\leq X}\lambda(n)n^{it}|^2dt +\int_N |\sum_{n\leq X}\lambda(n)n^{it}|^2dt$
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