Sarnak conjecture

  This is a note concentrate on the log average Sarnak conjecture, after the work of Matomaki and Raziwill on the estimate of multiplication function of short interval. Given a overview of the presented tools and method dealing with this conjectue.   1. Introduction Sarnak conjecture [1] assert that for any obersevable $latex {\{f(T^n(x_0))\}_{n=1}^{\infty}}&fg=000000$ come from a determination systems $latex {(T,X),T:X\rightarrow X}&fg=000000$, where $latex {h(T)=0}&fg=000…

I wish to establish the following estimate: Conjecture :(correlation of Mobius function and nil-sequences in short interval) $ \lambda(n)$ is the liouville function we wish the following estimate is true. $ \int_{0\leq x\leq X}|\sup_{f\in \Omega^m}\sum_{x\leq n\leq x+H}\lambda(n)e^{2\pi if(x)}|dx =o(XH)$. Where we have $ H\to \infty$ as $ x\to \infty$, $ \Omega^m=\{a_mx^m+a_{m-1}x^{m-1}+…+a_1x+a_0 | a_m,…,a_1,a_0\in [0,1]\}$ is a compact space. I do not know how to prove this but thi…

Bourgain-Sarnak-Ziegler定理可以视为Vingrodov均值定理的有限版本。

Cylinder map: Cylender map:这是一个动力系统$ \Theta=(T,T^2)$,$ T:T^2\longrightarrow T^2 $满足：\\ $ T(x)=x+\alpha,T(y)=cx+y+h(x)$ 因此 $ y_1(n)=T^{n}(x)=x+n\alpha,y_2(n)=T^n(y)=nx+\frac{n(n-1)}{2}\alpha+y+\sum_{n=1}^{N-1}h(x+i\alpha) $ 来自动力系统$ \Theta$中的可观测量是指$ \xi(n)=f(T^n(x))$,其中$ x\in T^2$,$f\in C(T^2)$. 由于Cylender map是零熵的，这个情形下Sarnak猜想成立等价于: $ S(N)=\sum_{n=1}^N\mu(n)\xi(n)=\sum{n=1}^N \mu(n)f(T^nx) $ 满足$ S(N)=o(N)$,由于$ f_{\lambda_1\lambda_2}=e^{2\pi i(\lambda_1 x+\lambda_2 y)}$是$ C(T^2)$的一组基，只需对$ f_{\lamb…