数论导引

Introduction

1 A Brief History of Prime

1.1 Euclid and Primes

1.2 Summing Over the Primes

1.3 Listing the Primes

1.4 Fermat Numbers

1.5 Primality Testing

1.6 Proving the Fundamental Theorem of Arithmetic

1.7 Euclid's Theorem Revisited

2 Diophantine Equations

2.1 Pythagoras

2.2 The Fundamental Theorem of Arithmetic in Other Contexts

2.3 Sums of Squares

2.4 Siegel's Theorem

2.5 Fermat, Catalan, and Euler

3 Quadratic Diophantine Equations

3.1 Quadratic Congruences

3.2 Euler's Criterion

3.3 The Quadratic Reciprocity Law

3.4 Quadratic Rings

3.5 Units in Z

3.6 Quadratic Forms

4 Recovering the Fundamental Theorem of Arithmetic

4.1 Crisis

4.2 An Ideal Solution

4.3 Fundamental Theorem of Arithmetic for Ideals

4.4 The Ideal Class Group

5 Elliptic Curves

5.1 Rational Points

5.2 The Congruent Number Problem

5.3 Explicit Formulas

5.4 Points of Order Eleven

5.5 Prime Values of Elliptic Divisibility Sequences

5.6 Ramanujan Numbers and the Taxicab Problem

6 Elliptic Functions

6.1 Elliptic Functions

6.2 Parametrizing an Elliptic Curve

6.3 Complex Torsion

6.4 Partial Proof of Theorem 6.5

7 Heights

7.1 Heights on Elliptic Curves

7.2 Mordell's Theorem

7.3 The Weak Mordell Theorem: Congruent Number Curve

7.4 The Parallelogram Law and the Canonical Height

7.5 Mahler Measure and the Naive Parallelogram Law

8 The Riemann Zeta Function

8.1 Euler's Summation Formula

8.2 Multiplicative Arithmetic Functions

8.3 Dirichlet Convolution

8.4 Euler Products

8.5 Uniform Convergence

8.6 The Zeta Function Is Analytic

8.7 Analytic Continuation of the Zeta Function

9 The Functional Equation of the Riemann Zeta Function

9.1 The Gamma Function

9.2 The Functional Equation

9.3 Fourier Analysis on Schwartz Spaces

9.4 Fourier Analysis of Periodic Functions

9.5 The Theta Function

9.6 The Gamma Function Revisited

10 Primes in an Arithmetic Progression

10.1 A New Method of Proof

10.2 Congruences Modulo 3

10.3 Characters of Finite Abelian Groups

10.4 Dirichlet Characters and L-Functions

10.5 Analytic Continuation and Abel's Summation Formula

10.6 Abel's Limit Theorem

11 Converging Streams

11.1 The Class Number Formula

11.2 The Dedekind Zeta Function

11.3 Proof of the Class Number Formula

11.4 The Sign of the Gauss Sum

11.5 The Conjectures of Birch and Swinnerton-Dyer

12 Computational Number Theory

12.1 Complexity of Arithmetic Computations

12.2 Public-key Cryptography

12.3 Primality Testing: Euclidean Algorithm

12.4 Primality Testing: Pseudoprimes

12.5 Carmichael Numbers

12.6 Probabilistic Primality Testing

12.7 The Agrawal-Kayal-Saxena Algorithm

12.8 Factorizing

12.9 Complexity of Arithmetic in Finite Fields

References

Index

《数论导引》是“国外数学名著系列”之一，从最初等的数论知识谈起，一直讲到解析数论、代数数论、椭圆曲线以及数论在密码理论中的应用等，涉及范围很广阔，而且内容并不肤浅。书中还有不少练习题，以及历史的评注等。可供数论及相关专业研究生、教师及科研人员等学习参考。

An Introduction to Number Theory provides an introduction to themain streams of number theory.Starting with the unique factorizationproperty of the integers， the theme of factorization is revisited severaltimes throughout the book to illustrate how the ideas handed downfrom Euclid continue to reverberate through the subject. In particular，the book shows how the Fundamental Theorem of Arithmetic， handeddown from antiquity， informs much of the teaching of modem numbertheory. The result is that number theory will be understood， not asa collection of tricks and isolated results， but as a coherent andinterconnected theory. A number of different approaches to numbertheory are presented， and the different streams in the book are broughttogether in a chapter that describes the class number formula forquadratic fields and the famous conjectures of Birch and Swinnerton-Dyer. The final chapter introduces some of the main ideas behindmodern computational number theory and its applications incryptography. Written for graduate and advanced undergraduatestudents 'of mathematics， this text will also appeal to students incognate subjects who wish to learn some of the big ideas in numbertheory.