数论导引
作者：[英国] 埃弗里斯特（Graham Everest）著
出版时间：2011
内容简介
《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。
目录
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
