《解析数论导论(英文版)》是一部为本科生提供学习数论的基本思想和技巧的教程，重点强调解析数论。前五章讲述可约性、收敛和算术函数等基本概念。紧下来的章节讲述序列中素数的狄利克莱定理、高斯和、二次剩余、狄利克莱级数和欧拉积及其在黎曼zeta函数和狄利克莱函数中的应用，并且引进了划分的概念。书中每章末都收集了大量练习。前十章，除去第一章，任何具备基本微积分知识的人都可以读懂；最后四章需要对复函数理论(包括复积分和留数积分)一定的了解。本书由(美)阿波斯托尔著。

Historical Introduction

Chapter 1

The Fundamental Theorem of Arithmetic

1.1 Introduction

1.2 Divisibility

1.3 Greatest common divisor

1.4 Prime numbers

1.5 The fundamental theorem of arithmetic

1.6 The series of reciprocals of the primes

1.7 The Euclidean algorithm

1.8 The greatest common divisor of more than two, numbers

 Exercises for Chapter 1

Chapter 2

Arithmetical Functions and Dirichlet Multiplication

2.1 Introduction

2.2 The M6bius function (n)

2.3 The Euler totient function (n)

2.4 A relation connecting and u

2.5 A product formula for (n)

2.6 The Dirichlet product of arithmetical functions

2.7 Dirichlet inverses and the M6bius inversion formula

2.8 The Mangoldt function A(n)

2.9 Muitiplicative functions

2.10 Multiplicative functions and Dirichlet multiplication

2.11 The inverse of a completely multiplicative function

2.12 Liouville's function)

2.13 The divisor functions a,(n)

2.14 Generalized convolutions

2.15 Formal power series

2.16 The Bell series of an arithmetical function

2.17 Bell series and Dirichlet multiplication

2.18 Derivatives of arithmetical functions

2.19 The Selberg identity

 Exercises for Chapter 2

Chapter 3

Averages of Arithmetical Functions

3.1 Introduction

3.2 The big oh notation. Asymptotic equality of functions

3.3 Euler's summation formula

3.4 Some elementary asymptotic formulas

3.5 The average order of din)

3.6 The average order of the divisor functions a,(n)

3.7 The average order of ~0(n)

3.8 An application to the distribution of lattice points visible from the origin

3.9 The average order of/4n) and of A(n)

3.10 The partial sums ofa Dirichlet product

3.11 Applications to pin) and A(n)

3.12 Another identity for the partial gums of a Dirichlet product

 Exercises for Chapter 3

Chapter 4

Some Elementary Theorems on the Distribution of Prime

Numbers

4.1 Introduction

4.2 Chebyshev's functions (x) and (x)

4.3 Relations connecting/x) and n(x)

4.4 Some equivalent forms of the prime number theorem

4.5 Inequalities for (n) and p,

4.6 Shapiro's Tauberian theorem

4.7 Applications of Shapiro's theorem

4.8 An asymptotic formula for the partial sums, (I/p)

4.9 The partial sums of the M6bius function 91

4.10 Brief sketch of an elementary proof of the prime number theorem

4.11 Selbcrg's asymptotic formula

 Exercises for Chapter 4

Chapter 5

Congruences

5.1 Definition and basic properties of congruences

5.2 Residue classes and complete residue systems

5.3 Linear congruences

Chapter 6

Finite Abelian Groups and Their Characters

Chapter 7

Dirichlet's Theorem on Primes in Arithmetic Progressions

Chapter 8

Periodic Arithmetical Functions and Gauss Sums

Chapter 9

Quadratic Residues and the Quadratic Reciprocity Law

Chapter 10

Primitive Roots

Chapter 11

Dirichlet Series and Euler Products

Chapter 12

The Functions (s) and L(s,x)

Chapter 13

Analytic Proof of the Prime Number Theorem