Analytic number theory

A pdf version is Dirichlet hyperbola method. 1. Introduction Theorem 1 $\sum_{1\leq n\leq x}d(n)=\sum_{1\leq n\leq x}[\frac{x}{n}]=xlogx+(2\gamma-1) x+O(\sqrt{x}) \ \ \ \ \ (1)$ Remark 1 I thought this problem initial 5 years ago, cost me several days to find a answer, I definitely get something without the argument of Dirchlet hyperbola method and which is weaker but morally the same camparable with the result get by Dirichlet hyperbola method. Remark 2 How to get the formula: $\sum_{1\leq n\le…

-1.Motivation- Large sieve: A Philosophy Reflecting a Large Group of Inequalities The large sieve is a philosophy that reflects a large group of inequalities which are very effective in controlling some linear sums or square sums of correlations of arithmetic functions. This idea could have originated in harmonic analysis, relying almost entirely on almost orthogonality. One fundamental example is the estimate of the quality: $$\sum_{n\leq x}|\Lambda(n)\overline{\chi(n)}|$$ One naive idea to con…

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…

An important theorem in Diophantine approximation is the theorem of Liuoville: **Liuoville Theorem** If x is a algebraic number of degree $latex n$ over the rational number then there exists a constant $latex c(x) > 0$ such that:$latex \left|x-{\frac {p}{q}}\right|>{\frac {c(x)}{q^{{n}}}}$ holds for every integer $latex p,q\in N^*$ where $latex q>0$. This theorem explain a phenomenon, the approximation of algebraic number by rational number could not be very well. Which was generated la…

The most important background of analytic number theory is the new understanding of the multiplication function on the shared interval. This result is established by Kaisa Matomäki & Maksym Radziwill, two very young and intelligent superstars. The main theorem in their article is: Theorem (Matomaki, Radziwill):As soon as $H \to \infty$ when $x \to \infty$, one has:$$\sum_{x\leq n\leq x+H}\lambda(n)= o(H)$$for almost all $x\sim X$. In my understanding of the result, the main strategy is: Step…

Vinogradov estimate is: $ |\sum_{n=1}^{N}e^{2\pi i\alpha P(n)}|\leq c_A\frac{N}{log^A N}$ For fix $ \alpha$ is irrational and $ \forall A>0 … (*)$. Assume $ deg(P)=n$, this could view as a effective uniformly distribute result of dynamic system:  $ ([0,1]^n,T)$, where $ T: x\to (A+B)x$, $ b$ is a nilpotent matrix, matrix $ A$ is identity but with a irrational number $ \alpha$ in the $ (n, n)$ elements. First approach we could easily to get a “uniform distribute on fiber&#8221…

this note will talk about the Ostrowski representation and approximation by continue fraction. As well-known,by the Weyl criterion,$latex \{n\alpha\}$ is uniformly distribution in $latex [0,1]$ iff $latex \alpha\in R-Q$. i.e. we have:$latex \forall 0\leq a\leq b\leq 1$,we have: $latex \lim_{N\to \infty}|\{1\leq n\leq N|\{n\alpha\}\in [a,b]\}|=(b-a)N+o(N)$. but this will not give the effective version.i.e. we do not the the more information about the decay of $latex o(N)$. we will give a approach…

There is a main obstacle to improve the kakeya conjecture,remain in dimension 3,and the  result established by Tomas Wolff in 1995 is almost the best result in $ R^3$ even until now.the result establish by Katz and Tao can be view as a corollary of Wolff’s X-ray estimate. For $ f\in L_{loc}^1(R^d)$,for $ 0<\delta<1$: $ f_{\delta}^*:P^{d-1}\longrightarrow R$.$ f_{\delta}^*(e)=\sup_{T}\frac{1}{|T|}\int_{T}|f|$. $ T$ is varise in all cylinders with length 1.radius $ \delta$.axis in…

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…