数论中的基本方法（影印版）|Graduate Texts in Mathematics|Melvyn B.Nathanson|世界图书出版公司

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数论中的基本方法（影印版）

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原书名：Elementary Methods in Number Theory	原出版社：Springer-Verlag
作者： Melvyn B.Nathanson	出版社：世界图书出版公司
译者：	丛书名： Graduate Texts in Mathematics
出版日期：2003-6-1	上架日期：2005-10-8
ISBN：7506259567	页数：版次：1-1
开本：24开	装帧：

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内容简介
This book, Elementary Methods in Number Theory, is divided into three parts.
Part I, "A first course in number theory," is a basic introduction to elementary number theory for undergraduate and graduate students with no previous knowledge of the subject. The only prerequisites are a little calculus and algebra, and the imagination and perseverance to follow a mathematical argument. The main topics are divisibility and congruences. We prove Gauss's law of quadratic reciprocity, and we determine the moduli for which primitive roots exist. There is an introduction to Fourier analysis on finite abelian groups, with applications to Gauss sums. A chapter is devoted to the abc conjecture, a simply stated but profound assertion about the relationship between the additive and multiplicative properties of integers that is a major unsolved problem in number theory.

目录
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Preface
Notation and conventions
I  A First Course in Number Theory
1  Divisibility and Primes
1.1  Division Algorithm
1.2  Greatest Common Divisors
1.3  The Euclidean Algorithm and Continued Fractions
1.4  The Fundamental Theorem of Arithmetic
1.5  Euclid's Theorem and the Sieve of Eratosthenes
1.6  A Linear Diophantine Equation
1.7  Notes
2  Congruences
2.1  The Ring of Congruence Classes
2.2  Linear Congruences
2.3  The Euler Phi Function
2.4  Chinese Remainder Theorem
2.5  Euler's Theorem and Fermat's Theorem
2.6  Pseudoprimes and Carmichael Numbers
2.7  Public Key Cryptography
2.8  Notes
3  Primitive Roots and Quadratic Reciprocity
3.1  Polynomials and Primitive Roots
3.2  Primitive Roots to Composite Moduli
3.3  Power Residues
3.4  Quadratic Residues
3.5  Quadratic Reciprocity Law
3.6  Quadratic Residues to Composite Moduli
3.7  Notes
4  Fourier Analysis on Finite Abelian Groups
4.1  The Structure of Finite Abelian Groups
4.2  Characters of Finite Abelian Groups
4.3  Elementary Fourier Analysis
4.4  Poisson Summation
4.5  Trace Formulae on Finite Abelian Groups
4.6  Gauss Sums and Quadratic Reciprocity
4.7  The Sign of the Gauss Sum
4.8  Notes
5  The abc Conjecture
5.1  Ideals and Radicals
5.2  Derivations
5.3  Mason's Theorem
5.4  The abc Conjecture
5.5  The Congruence abc Conjecture
5.6  Notes
II  Divisors and Primes in Multiplicative Number Theory
6  Arithmetic Functions
6.1  The Ring of Arithmetic Functions
6.2  Mean Values of Arithmetic Functions
6.3  The MSbius Function
6.4  Multiplicative Functions
6.5  The mean value of the Euler Phi Function
6.6  Notes
7  Divisor Functions
7.1  Divisors and Factorizations
7.2  A Theorem of Ramanujan
7.3  Sums of Divisors
7.4  Sums and Differences of Products
7.5  Sets of Multiples
7.6  Abundant Numbers
7.7  Notes
8  Prime Numbers
8.1  Chebyshev's Theorems
8.2  Mertens's Theorems
8.3  The Number of Prime Divisors of an Integer
8.4  Notes
9  The Prime Number Theorem
9.1  Generalized Von Mangoldt Functions
9.2  Selberg's Formulae
9.3  The Elementary Proof
9.4  Integers with k Prime Factors
9.5  Notes
10  Primes in Arithmetic Progressions
10.1  Dirichlet Characters
10.2  Dirichlet L-Functions
10.3  Primes Modulo 4
10.4  The Nonvanishing of L(1, X)
10.5  Notes
III  Three Problems in Additive Number Theory
11  Waring's Problem
11.1  Sums of Powers
11.2  Stable Bases
11.3  Shnirel'man's Theorem
11.4  Waring's Problem for Polynomials
11.5  Notes
12  Sums of Sequences of Polynomials
12.1  Sums and Differences of Weighted Sets
12.2  Linear and Quadratic Equations
12.3  An Upper Bound for Representations
12.4  Waring's Problem for Sequences of Polynomials
12.5  Notes
13  Liouville's Identity
13.1  A Miraculous Formula
13.2  Prime Numbers and Quadratic Forms
13.3  A Ternary Form
13.4  Proof of Liouville's Identity
13.5  Two Corollaries
13.6  Notes
14  Sums of an Even Number of Squares
14.1  Summary of Results
14.2  A Recursion Formula
14.3  Sums of Two Squares
14.4  Sums of Four Squares
14.5  Sums of Six Squares
14.6  Sums of Eight Squares
14.7  Sums of Ten Squares
14.8  Notes
15  Partition Asymptotics
15.1  The Size ofp(n)
15.2  Partition Functions for Finite Sets
15.3  Upper and Lower Bounds for logp(n)
15.4  Notes
16  An Inverse Theorem for Partitions
16.1  Density Determines Asymptotics
16.2  Asymptotics Determine Density
16.3  Abelian and Tauberian Theorems
16.4  Notes
References
Index

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