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# 数学代写| Existence of the Rubinstein{Sarnak distribution 数论代考

## 数论代考

The proof of Theorem $5.2 .2$ depends roughly on two ingredients:

• On the arithmetic side, we can represent the arithmetic random functions $\mathrm{N}_{\mathrm{X}}$ as combinations of $x \mapsto x^{i \gamma}$, where the $\gamma$ are ordinates of zeros of the L-functions modulo $q$;
• Once this is done, we observe that Kronecker’s Equidistribution Theorem (Theorem B.6.5) implies convergence in law for any function of this type.
There are some intermediate approximation steps involved, but the ideas are quite intuitive.

In this section, we always assume the validity of the Generalized Riemann Hypothesis modulo q, unless otherwise noted.
For a Dirichlet character $\chi$ modulo $q$, we define random variables $\psi_{\chi}$ on $\Omega_{\mathrm{X}}$ by
$$\psi_{\chi}(x)=\frac{1}{\sqrt{x}} \sum_{n \leqslant x} \Lambda(n) \chi(n)$$
for $x \in \Omega_{\mathrm{X}}$, where $\Lambda$ is the von Mangoldt function (see Section C.4, especially (C.6), for the definition of this function).

The next lemma is a key step to express $N_{X}$ in terms of Dirichlet characters. It looks first like standard harmonic analysis, but there is a subtle point in the proof that is crucial for the rest of the argument, and for the very existence of the Chebychev bias.
LEMMA 5.3.1. We have
$$\mathrm{N}{\mathrm{X}, q}=m{q}+\sum_{\chi(\bmod q)}^{} \psi_{\chi} \bar{\chi}+\mathrm{E}{\mathrm{X}, q}$$ where $\mathrm{E}{\mathrm{X}, q}$ converges to 0 in probability as $\mathrm{X} \rightarrow+\infty$. PROOF. By orthogonality of the Dirichlet characters modulo $q$ (see Proposition C.5.1), we have
$$\varphi(q) \pi(x ; q, a)=\sum_{\chi(\bmod q)} \overline{\chi(a)} \sum_{p \leqslant x} \chi(p)$$
hence
$$\frac{\log x}{\sqrt{x}}(\varphi(q) \pi(x ; q, a)-\pi(x))=\sum_{\chi(\bmod q)}^{ } \frac{}{\chi(a)} \frac{\log x}{\sqrt{x}} \sum_{p \leqslant x} \chi(p)+\mathrm{O}\left(\frac{\log x}{\sqrt{x}}\right)$$
for $x \geqslant 2$, where the error term accounts for primes $p$ dividing $q$ (for which the trivial character takes the value 0 instead of 1 ); in particular, the implied constant depends on $q$. We now need to connect the sum over primes, for a fixed character $\chi$, to $\psi_{\chi}$. Recall that the von Mangoldt functions differs little
multiplied by the logarithm function. The sum of this simpler function is the random variable defined by $\theta_{\chi}(x)=\frac{1}{\sqrt{x}} \sum_{p \leqslant x} \chi(p) \log (p)$
$\Omega_{\mathrm{x}}$. It is related $\theta_{\chi}(x)-\psi_{\chi}(x)=-\frac{1}{\sqrt{x}} \sum_{k \geqslant 2} \sum_{p^{k} \leqslant x} \chi\left(p^{k}\right) \log p=-\frac{1}{\sqrt{x}} \sum_{k \geqslant 2} \sum_{p^{k} \leqslant x} \chi(p)^{k} \log p .$
We can immediately see that the contribution of $k \geqslant 3$ is very small: since the exponent $k$ is at most of size $\log x$, and $|\chi(p)| \leqslant 1$ for all primes $p$, it is bounded by
where the implied constant is absolute.
For $k=2$, there are two cases. If $\chi^{2}$ is the trivial character then
$$\frac{1}{\sqrt{x}} \sum_{p \leqslant \sqrt{x}} \chi(p)^{2} \log p=\frac{1}{\sqrt{x}} \sum_{\substack{p \leqslant \sqrt{x} \ p \nmid q}} \log p=1+\mathrm{O}\left(\frac{1}{\log x}\right)$$
by a simple form of the Prime Number Theorem in arithmetic progressions (the Generalized Riemann Hypothesis would of course give a much better error term, but this is not needed here). If $\chi^{2}$ is non-trivial, then we have
$$\frac{1}{\sqrt{x}} \sum_{p \leqslant \sqrt{x}} \chi(p)^{2} \log p \ll \frac{1}{\log x}$$
for the same reason. Thus we have

• 在算术方面，我们可以将算术随机函数 $\mathrm{N}_{\mathrm{X}}$ 表示为 $x \mapsto x^{i \gamma}$ 的组合，其中 $\gamma$是 L 函数模 $q$ 的零点的纵坐标；
• 一旦完成，我们观察到 Kronecker 的均衡分布定理（定理 B.6.5）意味着任何此类函数的收敛性。
涉及一些中间近似步骤，但这些想法非常直观。

$$\psi_{\chi}(x)=\frac{1}{\sqrt{x}} \sum_{n \leqslant x} \Lambda(n) \chi(n)$$

$$\mathrm{N}{\mathrm{X}, q}=m{q}+\sum_{\chi(\bmod q)}^{*} \psi_{\chi} \bar{\chi}+\ mathrm{E}{\mathrm{X}, q}$$ 其中 $\mathrm{E}{\mathrm{X}, q}$ 收敛到 0 的概率为 $\mathrm{X} \rightarrow+\infty$。证明。通过 Dirichlet 字符模 $q$ 的正交性（见命题 C.5.1），我们有
$$\varphi(q) \pi(x ; q, a)=\sum_{\chi(\bmod q)} \overline{\chi(a)} \sum_{p \leqslant x} \chi(p)$$

$$\frac{\log x}{\sqrt{x}}(\varphi(q) \pi(x ; q, a)-\pi(x))=\sum_{\chi(\bmod q)}^{ *} \frac{}{\chi(a)} \frac{\log x}{\sqrt{x}} \sum_{p \leqslant x} \chi(p)+\mathrm{O}\left(\ frac{\log x}{\sqrt{x}}\right)$$

$\Omega_{\mathrm{x}}$。它是相关的 $\theta_{\chi}(x)-\psi_{\chi}(x)=-\frac{1}{\sqrt{x}} \sum_{k \geqslant 2} \sum_{p^ {k} \leqslant x} \chi\left(p^{k}\right) \log p=-\frac{1}{\sqrt{x}} \sum_{k \geqslant 2} \sum_{p^ {k} \leqslant x} \chi(p)^{k} \log p .$

$$\frac{1}{\sqrt{x}} \sum_{p \leqslant \sqrt{x}} \chi(p)^{2} \log p=\frac{1}{\sqrt{x}} \ sum_{\substack{p \leqslant \sqrt{x} \ p \nmid q}} \log p=1+\mathrm{O}\left(\frac{1}{\log x}\right)$$

$$\frac{1}{\sqrt{x}} \sum_{p \leqslant \sqrt{x}} \chi(p)^{2} \log p \ll \frac{1}{\log x}$$

## 数论代写

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## 编码理论代写

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## 复分析代考

(1) 提到复变函数 ，首先需要了解复数的基本性左和四则运算规则。怎么样计算复数的平方根， 极坐标与 $x y$ 坐标的转换，复数的模之类的。这些在高中的时候囸本上都会学过。
(2) 复变函数自然是在复平面上来研究问题，此时数学分析里面的求导数之尖的运算就会很自然的 引入到复平面里面，从而引出解析函数的定义。那/研究解析函数的性贡就是关楗所在。最关键的 地方就是所谓的Cauchy一Riemann公式，这个是判断一个函数是否是解析函数的关键所在。
(3) 明白解析函数的定义以及性质之后，就会把数学分析里面的曲线积分 $a$ 的概念引入复分析中， 定义几乎是一致的。在引入了闭曲线和曲线积分之后，就会有出现复分析中的重要的定理: Cauchy 积分公式。 这个是易分析的第一个重要定理。