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主 编 杨劲根
副主编 楼红卫 李振钱 郝群
编写人员（按汉语拼音为序）
陈超群 陈猛 东瑜昕 高威 郝群 刘东弟 吕志 童裕孙 王巨平 王泽军 徐晓津 杨劲根 应坚刚 张锦豪 张永前 周子翔 朱胜林
1. 序言
2. 非数学专业的数学教材
3.数学分析和泛函分析
4.单复变函数
5.多复变函数
6.代数
7.数论
8.代数几何
9.拓扑与微分几何
10.偏微分方程
11.概率论
12.计算数学
13.其他
14.附录
数论是数学中历史悠久但又有生命力的分支，也很有趣味。数论有分初等数论、代数数论、解析数论等。初等数论主要用初等的方法讨论整数的性质，如同余方程、不定方程、二次剩余等。
代数数论是讨论代数数的分支，要使用很深的代数工具。近年来，代数数论和代数几何合起来形成了一门称为算术几何的新分支，是非常艰深的。解析数论则是用数学分析和复变函数论来研究数论的问题，当今数学中第一号未解决问题黎曼猜想就属于解析数论的范围。
大学本科阶段学习一些数论是有用的，特别对学习抽象代数有很大帮助。研究生阶段一般只有数论专业的学生才学数论。
我们在这里介绍的数论教材大多是供数学专业本科生使用的。
书名：Elementary Methods in Number Theory
作者： Melvyn B.Nathanson
出版商： Springer-Verlag
页数：509
适用范围：大学数学系本科基础数学学生、数学专业研究生
预备知识：微积分
习题数量：大
习题难度： 中等，多数习题很容易
推荐强度：9.5
书评： 本书是Springer-Verlag出版的研究生系列教材中的一本，编号第195，2000年出版。全书分 为三个部分，第一部分介绍了初等数论的基本内容，整除性，同余，原根，Gauss二次互反律，有限 交换群上的Fourier分析，以及abc猜想的一个简单介绍。第二部分讨论了一些算术函数的性质， 给出了素数定理的初等证明。第三部分介绍了加法数论中的三个问题，即Waring问题， 正整数表为整数的平方和的问题，以及分拆函数的渐进估计的问题。 本书的一个特点是给出了许多深刻的数论定理的初等证明， 比如，Selberg的素数定理的初等证明，Linnik关于Waring问题初等证明， 一个整数表示为偶数个整数的平方和的个数的Liouville方法，以及Erdos关于分拆函数的渐进估计的结果。 事实上，本书的所有的证明都只使用了初等的方法，不涉及解析方法以及其他的高等方法， 因此本书也是一本很好的大学生数论教材。本书第一部分和第二部分作为大学生一个学期的课程是合适的。 （王巨平）
国外评论摘选
1) Every serious student of number theory should have this classic book on their shelf. Even though only elementary calculus and abstract algebra are used, a certain mathematical maturity is required. I feel the book is strongest in the area of elementary --not necessarily easy though -- analytic number theory (Hardy was a world class expert in analytic number theory). An elementary, but difficult proof of the Prime number Theorem using Selberg's Theorem is thoroughly covered in chapter 22.
While modern results in the area of algorithmic number theory are not presented nor is a systematic presentation of number theory given (it is not a textbook), it contains a flavor, inspiration and feel that is completely unique. It covers more disparate topics in number theory than any other n.t. book I know of. The fundamental results in classical, algebraic, additive, geometric, and analytic number theory are all covered. A beautifully written book.
Other recommended books on number theory in increasing order of difficulty:
1) Elementary Number Theory, By David Burton, Third Edition. Covers classical number theory. Suitable for an upper level undergraduate course. Primarily intended as a textbook for a one semester number theory course. No abstract algebra required for this book. Not a gem of a book like Davenport's The Higher Arithmetic, but a great book to seriously start learning number theory.
2) The Queen of Mathematics, by Jay Goldman. A historically motivated guide to number theory. A very clearly written book that covers number theory at a graduate or advanced undergraduate level. Covers much of the material in Gauss's Disquisitiones, but without all the detail. The book covers elementary number theory, binary quadratic forms, cyclotomy, Gaussian integers, quadratic fields, ideals, algebraic curves, rational points on elliptic curves, geometry of numbers, and introduces p-adic numbers. Only a slight bit of analytic number theory is covered. The best book in my opinion to start learning algebraic number theory. Wonderfully fills the otherwise troublesome gap between undergraduate and graduate level number theory.
Full of historical information hard to find elsewhere, very well researched. To cover all the material in this book would likely take two semesters, though most of the important material could be covered in one semester. Requires a background in abstract algebra (undergraduate level), and a little advanced calculus. Some complex analysis for sections 19.7 and 19.8 would be helpful, but not at all a requirement. The author recommends Harold Davenport's The Higher Arithmetic, as a companion volume for the first 12 chapters; according to Goldman it is a gem of a book.
3) Additive Number Theory, by Melvyn Nathanson. Graduate level text in additive number theory, covers the classical bases. This book is the first comprehensive treatment of the subject in 40 years. Some highlights: 1) Chen's theorem that every sufficiently large even integer is the sum of a prime and a number that is either prime or the product of two primes. 2) Brun's sieve for upper bound on the number of twin primes. 3) Vinogradov's simplification of the Hardy, Littlewood, and Ramanujan's circle method.
2) My initial reaction through the first chapters was one of embarrassment at my lack of understanding. I could not believe a book, hailed by so many as a standard and essential resource, could be so much out of my reach. Then, amid the last page or so of chapter 1 I had an epiphany. The book, from that point on, was completely clear and logical while retaining an extraordinary amount of breadth in coverage. Add my staunch support and recommendation to the long list of kudos that this book has accrued. There are, to my knowledge, no better books for the beginning student of number theory. If you have any interest whatsoever in the theory of numbers, this book is essential.
书名：A course in arithmetic
作者： J.-P. Serre
出版商： Springer Verlag (1973) ISBN 0-387-90041-1
页数：113
适用范围：大学数学系本科基础数学高年级或研究生低年级教材
预备知识：抽象代数,复分析
习题数量：很少
习题难度： 较大
推荐强度：10
书评： 法国大数学家，菲尔滋奖和阿贝尔奖获得者 Serre 写过不少短小精悍的小册子，大部分从他亲自所讲授的课程的讲稿整理而成。 本书是他非常有代表性的本科生高年级的数论教材，曾在西方评为某年度世界最佳数学教材。
本书并不是数论的系统教程，作者选择数论中三个重要专题扼要叙述了它们的内容和方法，这三个专题是：二次型、素数的 Dirichlet 定理 和模形式。读者可以化较少的时间学到一些近代数论的知识。最令读者欣赏的是定理的证明将大数学家的技巧展现得淋漓尽致，阅读中不禁 拍案叫绝。非定型幺模偶整格的分类定理非常漂亮，但其完整的证明在很多代数教科书中很难找到，本人所知道的就是本书以及 Milnor 和 Husemoeller 写的 Symmetric bilinear form 一书中的证明。这两本书都是70年代出版的，经过这两位菲尔滋奖得主之手的证明已经很难再 作改进，因此后人写的书大多只是引用而不再重写了。
具备抽象代数的知识就可以读懂前半本书，后一半需要复分析的准备知识。由于叙述简洁，习题数量少，作为教材使用会有一定困难。 作为自学的参考书对读者的数学素养也有较高的要求。 （杨劲根）
国外评论摘选
1) The book is divided into two parts -- algebraic and analytic. I've only worked through the analytic part. Anything by Serre is worth its weight in gold and this book is no exception; everything Serre covers is of the utmost importance. But Serre's style is extremely condensed and spare, and he makes no concessions to the reader in terms of motivation or examples. I can't digest more than half a page of Serre a day; however if one wants to understand the structure of a theory, Serre is ideal. I worked through A Course in Arithmetic over a decade back. As I recall I covered Riemann's zeta function and the Prime Number Theorem, the proof of Dirichlet's theorem on primes in arithmetical progressions using group characters in the context of arithmetical functions, and some of the basic theory of modular functions. All of this material is also covered in Apostol's two books on analytic number theory (Introduction to Analytic Number Theory, and Dirichlet Series and Modular Functions in Number Theory); Apostol goes further than Serre in the analytic part -- which is only to be expected since he is devoting two whole texts to the subject.
2) Serre's work could best be summarized in one word - Elegance. The book comprises of two distinct parts. The first one is the 'algebraic' part. Serre's goal in this section is to give a complete classification of the quadratic forms over the rationals. As preliminaries to reaching this goal, he introduces the reader to quadratic reciprocity, $p-$adic fields and the Hilbert Symbol. After these three, he spends the next chapter detailing the properties of quadratic forms over ${\mathbb Q$ and ${\mathbb Q_p$ (the $p-$adic field). The reason to work over ${\mathbb Q_p$ is the Hasse-Minkowski Theorem (which says that if you have a quadratic form, it has solutions in Q if and only if it has solutions in ${\mathbb Q_p$). Using Hensels Lemma, checking for solutions in ${\mathbb Q_p$ is (almost) as easy as checking for solutions in Z/pZ. After doing that, he spends yet another chapter talking about the quadratic forms over the integers. (Note: the classification goal is already achieved in previous chapter). The second half of the book is the 'analytic' one. The first chapter in this section gives a complete proof of Dirichlet's theorem while the second one studies the properties of modular forms (these are good!) Due to the extreme elegance, the book is sometimes hard to read. This might sound like a paradox, but it's not and I'll explain why. The book takes some effort to read because it's terse and it often takes a while to figure out why something is 'obvious'. However, once you see it all, you'll realize that a great mind was guiding you through the pursuit. The choice of topics is just right to achieve the goals that the author sets out for himself. Also, I'd rather think for myself and read a smaller book than be given a huge fat tome where the author details his own thought process. This book was my first foray into number theory and I absolutely enjoyed it. If you're considering reading it, I wish you joy in your pursuits.
书名：Introduction to Analytic Number Theory
作者： Tom Apostol
出版商： Springer Verlag (1976) ISBN 0-387-90163-9
页数：328
适用范围：大学数学系本科数论教材
预备知识：微积分,复分析
习题数量：大
习题难度：一般
推荐强度：9
书评： 这是一本非常受欢迎的数论入门教材，写得极其清楚仔细而又不烦琐。虽然书名是解析数论，事实上也包括了初等数论。 由于书的自封性能好，习题又经过精心挑选，适合于大学低年级的数论教材。
本书由于其良好声誉而多次再版，被选入 Springer 的 UTM 系列。同一作者的微积分教材（见本书的另一篇书评）也有好口碑。 （杨劲根）
国外评论摘选
1) I think that there will be little harm if the title of the book is changed to 'Introduction to elementary number theory' instead. The author presumes that the reader has not any knowledge of number theory. As a result, materials like congruence equation, primitive roots, and quadratic reciprocity are included. Of course as the title indicates, the book focusses more on the analytic aspect. The first 2 chapters are on arithmetic functions, asymptotic formulas for averaging sums, using elementary methods like Euler-Maclaurin formula .This lay down the foundation for further discussion in later chapters, where complex analysis is involved in the investigation. Then the author explain congruence in chapter 4 and 5. Chapter 6 introduce the important concept of character. Since the purpose of this chapter is to prepare for the proof of Dirichlet's theorem and introduction of Gauss sums, the character theory is developed just to the point which is all that's needed. ( i.e. the orthogonal relation). Chapter 7 culminates on the elementary proof on Dirichlet's theorem on primes in arithmetic progression. The proof still uses $L-$function of course, but the estimates, like the non-vanishing of $L(1)$ , are completely elementary and is based only on the first 2 chapters. The author then introduce primitve roots to further the theory of Dirichlet characters. Gauss sums can then be introduced. 2 proofs of quadratic reciprocity using Gauss sums are offered. The complete analytic proof, using contour integration to evaluate explicitly the quadratic Gauss sums, is a marvellous illustration of how truth about integers can be obtained by crossing into the complex domains. The book then turns in to the analyic aspect. General Dirichlet series, followed by the Riemann zeta function, L function ,are introduced. It's shown that the $L-$functions have meremorphic continuation to the whole complex plane by establishing the functional equation $L(s)=$ elementary factor $* L(1-s).$ The reader should be familiar with residue calculus to read this part. Chapter 13 may be a high point of this book, where the Prime Number Theorem is proved. Arguably, it's the Prime Number Theorem which stimulate much of the theory of complex analysis and analyic number theory. As Riemann first pointed out, the Prime Number Theorem can be proved by expressing the prime counting function as a contour integral of the Riemann zeta function, then estimate the various contours. The proof given in this book , although not exactly that envisaged by Riemann , is a variant that run quite smoothly. As is well known , a key point is that one can move the contour to the line $Re(s)=1,$ and to do this one have to verify that $\zeta(s)$ does not vanish on $Re(s)=1.$ The proof , due to de la vale-Poussin, is a clever application of a trigonometric identity. Unfortunately, the method does not allow one penetrate into the region $0<1,$ where the distribution of zeroes in this region contain the information about the flunctuation of $\Pi(x)$ around $x/\log x.$ The famous Riemann Hypothesis states that the only zeroes in this region lis on the line $Re(s)=1/2.$ After more than 100 years, although the Riemann Hpothesis has natural generalisation to number fields, neither of these RH is proven, which indicates the difficulties of this problem. Recently some new directions, related to quantum statistical mechanics, has been connected with this old problem. If the RH is proven, then the set of prime numbers , although looks completely random locally ( like the occurences of twin primes), is governed by clear-cut laws on the large after all. The last Chapter is of quite differnt flavour, the so-called additive number theory. Here the author only focusses on the simplest partition function ---the unrestricted partition. However interesting phenomeon occur already at this level. The first result is Euler's pentagonal number theorem, which leads to a simple recursion formula for the partition function p(n). 3 proofs are given. The most beautiful one is no doubt a combinatorial proof due to Franklin. The third proof is through establishing the Jacobi triple product identity, which leads to lots of identites besides Euler's pentagonal number theorem. Jacobi's original proof uses his theory of theta functions, but it turns out that power series manipulaion is all that's needed. The book ends with an indication of deeper aspect of partition theory--- Ramanujan's remarkable congrence and identities ( the simplest one being $p(5m+4)= 0 \pmod{5$ ). To prove these mysterious identites, the naturalway is to plow through the theory of modular functions, which Ramanujan had left lots more theorem ( unfortunately most without proof). However an elementary proof of one these identites is outlined in the exercises. This book is well written, with enough exercises to balance the main text. Not bad for just an 'introduction'.
2) This book has excellent exercises at the end of each chapter. The exercises are interesting and challenging and supplement the main text by showing additional consequences and alternate approaches. The book covers a mixture of elementary and analytic number theory, and assumes no prior knowledge of number theory. Analytic ideas are introduced early, wherever they are appropriate. The exposition is very clear and complete. Some novel features include: three chapters on arithmetic functions and their averages (including a simple Tauberian theorem due to Shapiro); Polya's inequality for character sums; and an evaluation of Gaussian sums (by contour integration), used in one proof of quadratic reciprocity.