Sum-product phenomenon
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Log average 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…
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The correlation of Mobius function and nil-sequences in short interval
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…
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Fractional uncertain principle
semyon dyatlov的一篇文章 semyon dyatlov的文章https://arxiv.org/pdf/1710.05430.pdf,用fractional uncertainly priciple导出了hyperbolic surface上测地线诱导的zeta函数在$latex Re(s)>1-\epsilon$只有有限个零点。 就我的理解,这件事情至少和3个事情有关系, 1.p-adic上的黎曼猜想,因为这篇文章的证明强烈依赖于markov性质,这和p adic的结构也很像,有可能可以利用p adic猜想的证明思路继续做一部分。 2.billiard的传播子,但是这里不一样,文章中的 Schottky groups本质上是对于算子的逆写成一种级数形式其中级数由Schottky group生成,但是对于billiard传播子的情况所有的涉及的热核或者波核的paramatrix不仅仅具备markov性质,起主导作用的却是某种需要X-ray估计的性质,级数和并不是对全空间求而是某种截断了的子空间里面,所以比这个证明要难。建立起billiar…