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• Statistical Inference 统计推断
• Statistical Computing 统计计算
• Advanced Probability Theory 高等概率论
• Advanced Mathematical Statistics 高等数理统计学
• (Generalized) Linear Models 广义线性模型
• Statistical Machine Learning 统计机器学习
• Longitudinal Data Analysis 纵向数据分析
• Foundations of Data Science 数据科学基础

## 数学代写|数论作业代写number theory代考|THE PRIME NUMBER THEOREM

For any real number $x$, we denote by $\pi(x)$ the number of primes less than or equal to $x$. For example, $\pi(1)=0$ and $\pi(10)=4$ and $\pi(100)=\pi(100.5)=25$. The symbol $\pi$ is customary and is used because it is the Greek equivalent of ” $\mathrm{p}$ “, the first letter of the word “prime” – it has, of course, nothing to do with the trigonometric constant $\pi$. The following result, which gives an estimate for $\pi(x)$ in terms of elementary functions, was proved independently and more or less simultaneously in 1896 by Hadamard [27] and de la Vallée Poussin [65]. A proof is given by Hardy and Wright [29].

Theorem 3.37. The Prime Number Theorem. The number of primes not exceeding $x$ satisfies
$$\frac{\pi(x)}{x / \log x} \rightarrow 1 \quad \text { as } x \rightarrow \infty$$
where log denotes the natural (base e) logarithm.

We may use the Prime Number Theorem to estimate the least common multiple of the first $n$ positive integers. Given a prime $p$ and a positive integer $n$, there is a unique non-negative integer $\alpha$ such that $p^{\alpha} \leq n<p^{\alpha+1}$; the least common multiple will be the product of all these powers $p^{\alpha}$. Therefore
$$L_{n}=\operatorname{lcm}(1,2, \ldots, n)=\prod_{\substack{p \leq n \ p \text { prime }}} p^{\alpha} \leq \prod_{\substack{p \leq n \ p \text { prime }}} n=n^{\pi(n)}$$
the last equality being true because the product consists of $\pi(n)$ equal factors. Let $c$ be a constant greater than $e$. It follows from the Prime Number Theorem that if $n$ is sufficiently large then
$$\frac{\pi(n)}{n / \log n}<\log c$$
and hence
$$L_{n} \leq n^{\pi(n)}=e^{\pi(n) \log n}<c^{n} .$$
In section $3.4$ we chose $c=3$ to keep things simple. We could have taken a slightly smaller value, which would have resulted in a slightly larger value for $s$; however, this would have made no significant difference to the result.

Comment. In fact, it can be shown [55] that the maximum value of $\left(L_{n}\right)^{1 / n}$ occurs when $n=113$. Therefore, if we take
$$c=(\operatorname{lcm}(1,2, \ldots, 113))^{1 / 113}=2.8258821394 \cdots,$$
then we have $L_{n} \leq c^{n}$ for all $n$, and not just for all sufficiently large $n$.

## 数学代写|数论作业代写number theory代考|DEFINITION AND BASIC PROPERTIES

Definition 4.1. A finite or infinite expression of the form
$$a_{0}+\frac{b_{1}}{a_{1}+\frac{b_{2}}{a_{2}+\frac{b_{3}}{a_{3}+\cdots}}}$$
is called a continued fraction. A simple continued fraction is one in which every $b_{k}$ is 1 , every $a_{k}$ is an integer, and every $a_{k}$ except possibly $a_{0}$ is positive. For a (finite or infinite) simple continued fraction we shall also use the notations
$$a_{0}+\frac{1}{a_{1}+} \frac{1}{a_{2}+} \frac{1}{a_{3}+\ldots} \quad \text { and } \quad\left[a_{0}, a_{1}, a_{2}, a_{3}, \ldots\right] .$$
A finite simple continued fraction is said to represent the number obtained by performing the arithmetic in the obvious way; an infinite simple continued fraction $\left[a_{0}, a_{1}, a_{2}, a_{3}, \ldots\right]$ represents the real number $\alpha$ if
$$\alpha=\lim {n \rightarrow \infty}\left[a{0}, a_{1}, a_{2}, a_{3}, \ldots, a_{n}\right] .$$

Let $k \in \mathbb{N}$. The integer $a_{k}$ is called the $k$ th partial quotient of the continued fraction $\left[a_{0}, a_{1}, a_{2}, \ldots\right]$, or of the number $\alpha$ it represents; the continued fraction $\alpha_{k}=\left[a_{k}, a_{k+1}, a_{k+2}, \ldots\right]$ is the kth complete quotient of $\alpha$; and the continued fraction $\left[a_{0}, a_{1}, \ldots, a_{k}\right]$ is the kth convergent to $\alpha$.

Note that a convergent, being defined as a finite continued fraction, is always a rational number. Henceforth we shall blur the distinction between a continued fraction and the number represented by the continued fraction. We shall use such language as, for example, “the continued fraction $\alpha=\left[a_{0}, a_{1}, a_{2}, \ldots\right]^{\text {” }}$ instead of saying more precisely, “the continued fraction $\left[a_{0}, a_{1}, a_{2}, \ldots\right]$ which represents the number $\alpha \geqslant$

Continued fractions, their convergents, partial quotients and complete quotients have many fascinating properties.

## 数学代写|数论作业代写number theory代考|THE PRIME NUMBER THEOREM

$$\frac{\pi(x)}{x / \log x} \rightarrow 1 \quad \text { as } x \rightarrow \infty$$

$$L_{n}=\operatorname{lcm}(1,2, \ldots, n)=\prod_{p \leq n p \text { prime }} p^{\alpha} \leq \prod_{p \leq n p \text { prime }} n=n^{\pi(n)}$$

$$\frac{\pi(n)}{n / \log n}<\log c$$

$$L_{n} \leq n^{\pi(n)}=e^{\pi(n) \log n}<c^{n}$$

$$c=(\operatorname{lcm}(1,2, \ldots, 113))^{1 / 113}=2.8258821394 \cdots,$$

## 数学代写|数论作业代写number theory代考|DEFINITION AND BASIC PROPERTIES

$$a_{0}+\frac{b_{1}}{a_{1}+\frac{b_{2}}{a_{2}+\frac{b_{3}}{a_{3}+\cdots}}}$$

$$a_{0}+\frac{1}{a_{1}+} \frac{1}{a_{2}+} \frac{1}{a_{3}+\ldots} \quad \text { and } \quad\left[a_{0}, a_{1}, a_{2}, a_{3}, \ldots\right]$$

$$\alpha=\lim n \rightarrow \infty\left[a 0, a_{1}, a_{2}, a_{3}, \ldots, a_{n}\right] .$$

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MATLAB 是一种用于技术计算的高性能语言。它将计算、可视化和编程集成在一个易于使用的环境中，其中问题和解决方案以熟悉的数学符号表示。典型用途包括：数学和计算算法开发建模、仿真和原型制作数据分析、探索和可视化科学和工程图形应用程序开发，包括图形用户界面构建MATLAB 是一个交互式系统，其基本数据元素是一个不需要维度的数组。这使您可以解决许多技术计算问题，尤其是那些具有矩阵和向量公式的问题，而只需用 C 或 Fortran 等标量非交互式语言编写程序所需的时间的一小部分。MATLAB 名称代表矩阵实验室。MATLAB 最初的编写目的是提供对由 LINPACK 和 EISPACK 项目开发的矩阵软件的轻松访问，这两个项目共同代表了矩阵计算软件的最新技术。MATLAB 经过多年的发展，得到了许多用户的投入。在大学环境中，它是数学、工程和科学入门和高级课程的标准教学工具。在工业领域，MATLAB 是高效研究、开发和分析的首选工具。MATLAB 具有一系列称为工具箱的特定于应用程序的解决方案。对于大多数 MATLAB 用户来说非常重要，工具箱允许您学习应用专业技术。工具箱是 MATLAB 函数（M 文件）的综合集合，可扩展 MATLAB 环境以解决特定类别的问题。可用工具箱的领域包括信号处理、控制系统、神经网络、模糊逻辑、小波、仿真等。

assignmentutor™您的专属作业导师
assignmentutor™您的专属作业导师