Number theory is not a simple subject. This subject is called number theory in China. Many students often find it not difficult when learning linear algebra or analysis (advance calculus), but they find number theory to be difficult. This is because Did not find the right way to learn number theory, UpriviateTA has a series of teachers who are very good at number theory, who can ensure that you get satisfactory results in the process of learning number theory.

number theory不算是一门简单的学科，这门学科在中国叫做数论，经常有很多学生在学linear algebra或者analysis(advance calculus)的时候觉得并不困难，但是却觉得数论number theory很难，这是因为没有找到正确的方法学习number theory，UpriviateTA有一系列非常擅长number theory的老师，可以确保您在学习数论的过程中取得满意的成绩。

Number Theory is the Area called by Carl Friedrich Gauss “the queen of mathematics”. One must give
Gauss a considerable amount of credit here. Paranthetically, in 1796 at the age of
nineteen, Gauss decided to dedicate his life to mathematics after he has shown how
a regular polygon of seventeen sides can be constructed with a straight edge and a
collapsible compass.

We end this introduction with two examples of such striking facts in number theory
whose statemets are nevertheless easy enough to understand. The first is an exercise

$$2^{32}+1=\left(2^{9}+2^{7}+1\right)\left(2^{23}-2^{21}+2^{19}-2^{17}+2^{14}-2^{9}-2^{7}+1\right)$$

which, for convincing one of its validity, it requires just a little algebra. Interesting enough is the fact that it shows that $2^{2^{n}}+1$ is not a prime number for all $n \in \mathbb{N}$ (as Fermat predicted and gave Euler’s the opportunity to show off with his calculatorial powers by giving this counterexample).

The second is a theorem of Hurwitz from 1891:

\begin{thm}
Theorem 0.0.1. For every irrational number $x$ there exist infinitely many pairs of integers $p$ and $q,$ such that
$$\left|x-\frac{p}{q}\right| \leq \frac{1}{\sqrt{5} q^{2}}$$
The constant $\frac{1}{\sqrt{5}}$ is the best possible in the sense that if we replace it by something smaller, say $C^{0}>0,$ then there are infinitely many irrationals $x,$ for which only finitely many pairs of integers $p$ and $q$ satisfy
$$\left|x-\frac{p}{q}\right| \leq \frac{C}{q^{2}}$$
\end{thm}

Huwritz’s theorem is a very interesting theorem. An important point is that this theorem is sharp. $\frac{\sqrt{5}-1}{2}$ is an example of making this theorem equal, the proof of this theorem The idea is to analyze the proof process of Dirichlet’s theorem in more detail, and then do some classification discussions.

Finally, number theory abounds in old an new conjectures but one can come up easily with his own. A good reference for lots and lots of interesting and dramatic conjectures or facts in this area is R. Guy, Unsolved Problems in Number Theory. One of the million dollar conjectures or millennium problems is at the intersection of number theory and complex analysis. It is known as the Riemann Hypothesis. A reformulation of it is to show that ( A. Granville and G. Martin, Prime number races, American Mathematical
Monthly, vol 113, No 1, 2006.)

$\mid \int_{1}^{x} \frac{1}{\ln t} d t-\#\{p \mid p$ is prime such that $p \leq x\} \mid \leq \sqrt{x} \ln x,$ for $x \geq 3$.
Let us conclude with the observation that the topics of number theory are basically at the heart and very good introductions to other, more abstract and technical, branches of mathematics.

##### For the prime number theorem, many people have conducted extensive research on him. The first person to get a result is Dirichlet (not to solve the prime number theorem, but to promote it!):

Dirichlet’s remark from the first paper is extracted and translated on page 98 of The Development of Prime Number Theory by Narkiewicz. So this has not passed completely unnoticed. Narkiewicz remarks that Dirichlet believed that his analytic methods would enable him to prove Legendre’s conjecture, and that Dirichlet never returned to the problem.
Dirichlet remained interested in the asymptotic growth laws (“Asymptotische Gesetze”) of arithmetic functions for the rest of his life, as seen from his 1849 paper with the estimate

$$\sum_{n \leq x} d(n)=x \log (x)+(2 \gamma-1) x+O\left(x^{1 / 2}\right)$$
and a couple of other estimates, and a letter of 1858 to Kronecker reprinted in Dirichlet’s Werke, where he mentions having obtained a substantial improvement of the error term $O\left(x^{1 / 2}\right)$ by a new method.

Since Dirichlet demonstrably did not lose interest in such questions, and never returned to the PNT in print, it seems reasonable to believe that he discovered that his real-variable method would not yield the PNT.

##### Toronto MAT315H1代写请认准UpriviateTA
数论中的基本主题：算术函数；余类模m上的多项式，余类模m上的字符；二次互易定律，数字表示为平方和。

Elementary topics in number theory: arithmetic functions; polynomials over the residue classes modulo m, characters on the residue classes modulo m; quadratic reciprocity law, representation of numbers as sums of squares.

##### Berkeley MAT115代写请认准UpriviateTA
可除，同余，数值函数，素数理论。选择的主题：丢番图分析，连续分数，分区，二次场，渐近分布，加性问题。

Divisibility, congruences, numerical functions, theory of primes. Topics selected: Diophantine analysis, continued fractions, partitions, quadratic fields, asymptotic distributions, additive problems.

##### UCSD Math 204A代写请认准UpriviateTA
这是三门课程系列中的第一门，这是对代数和解析数论的入门。 A部分将讨论数字字段的基本属性：它们的整数环，唯一因式分解及其失败，类编号，狄利克雷单位定理，素数分裂，环原子场等。也将重点放在计算工具上，特别是SageMath和LMFDB。 （在2021年冬季，我将教授数学204B，它将涵盖更高级的主题。在2021年春季，克劳斯·索伦森将教授数学204C。）

This is the first in a series of three courses, which is an introduction to algebraic and analytic number theory. Part A will treat the basic properties of number fields: their rings of integers, unique factorization and its failure, class numbers, the Dirichlet unit theorem, splitting of primes, cyclotomic fields, and more. There will also be an emphasis on computational tools, particularly SageMath and the LMFDB. (In winter 2021 I will teach Math 204B, which will cover more advanced topics. In spring 2021, Claus Sorensen will teach Math 204C.)

Due to the COVID-19 pandemic and UCSD campus regulations, this course will be offered in a fully remote format. Lectures will be delivered live via Zoom, and also recorded for asynchronous viewing. Office hours will be held via Zoom; I also plan to offer in-person office hours to the extent possible. (If you were planning on using this course as an in-person course for immigration reasons, contact me for guidance.)

number theory代写请认准UpriviateTA. UpriviateTA为您的留学生涯保驾护航。