Book:Tom M. Apostol/Introduction to Analytic Number Theory

Tom M. Apostol: Introduction to Analytic Number Theory

Published $\text {1976}$, Springer-Verlag

Subject Matter

• Analytic Number Theory

Contents

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

Chapter 2. Arithmetical Functions and Dirichlet Multiplication

2.1 Introduction

2.2 The Mobius function $\map \mu n$

2.3 The Euler totient function $\map \phi n$

2.4 A relation connecting $\mu$ and $\phi$

2.5 A product formula for $\map \phi n$

2.6 The Dirichlet product of arithmetical functions

2.7 Dirichlet inverses and the Mobius inversion formula

2.8 The Mangoldt function $\map \Lambda n$

2.9 Multiplicative functions

2.10 Multiplicative functions and Dirichlet multiplication

2.11 The inverse of a completely multiplicative function

2.12 Liouville's function $\map \lambda n$

2.13 The divisor functions $\map {\sigma_x} n$

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

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 $\map d n$

3.6 The average order of the divisor functions $\map {\sigma_x} n$

3.7 The average order of $\map \phi n$

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

3.9 The average order of $\map \mu n$ and of $\map \Lambda n$

3.10 The partial sums of a Dirichlet product

3.11 Applications to $\map \mu n$ and $\map \Lambda n$

3.12 Another identity for the partial sums of a Dirichlet product

Chapter 4. Some Elementary Theorems on the Distribution of Prime Numbers

4.1 Introduction

4.2 Chebyshev's functions $\map \psi x$ and $\map \vartheta x$

4.3 Relations connecting $\map \vartheta x$ and $\map \pi x$

4.4 Some equivalent forms of the prime number theorem

4.5 Inequalities for $\map \pi n$ and $p_n$

4.6 Shapiro's Tauberian theorem

4.7 Applications of Shapiro's theorem

4.8 An asymptotic formula for the partial sums $\sum_{p \mathop \le x} 1 / p$

4.9 The partial sums of the Mobius function

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

4.11 Selberg's asymptotic formula

Chapter 5. Congruences

5.1 Definition and basic properties of congruences

5.2 Residue classes and complete residue systems

5.3 Linear congruences

5.4 Reduced residue systems and the Euler-Fermat theorem

5.5 Polynomial congruences modulo p. Lagrange's theorem

5.6 Applications of Lagrange's theorem

5.7 Simultaneous linear congruences. The Chinese remainder theorem

5.8 Applications of the Chinese remainder theorem

5.9 Polynomial congruences with prime power moduli

5.10 The principle of cross-classification

5.11 A decomposition property of reduced residue systems

Chapter 6. Finite Abelian Groups and Their Characters

6.1 Definitions

6.2 Examples of groups and subgroups

6.3 Elementary properties of groups

6.4 Construction of subgroups

6.5 Characters of finite abelian groups

6.6 The character group

6.7 The orthogonality relations for characters

6.8 Dirichlet characters

6.9 Sums involving Dirichlet characters

6.10 The nonvanishing of $\map L {1, \chi}$ for real nonprincipal $\chi$

Chapter 7. Dirichlet's Theorem on Primes in Arithmetic Progressions

7.1 Introduction

7.2 Dirichlet's theorem for primes of the form $4 n — 1$ and $4 n + 1$

7.3 The plan of the proof of Dirichlet's theorem

7.4 Proof of Lemma 7.4

7.5 Proof of Lemma 7.5

7.6 Proof of Lemma 7.6

7.7 Proof of Lemma 7.8

7.8 Proof of Lemma 7.7

7.9 Distribution of primes in arithmetic progressions

Chapter 8. Periodic Arithmetical Functions and Gauss Sums

8.1 Functions periodic modulo $k$

8.2 Existence of finite Fourier series for periodic arithmetical functions

8.3 Ramanujan's sum and generalizations

8.4 Multiplicative properties of the sums $\map {s_k} n$

8.5 Gauss sums associated with Dirichlet characters

8.6 Dirichlet characters with nonvanishing Gauss sums

8.7 Induced moduli and primitive characters

8.8 Further properties of induced moduli

8.9 The conductor of a character

8.10 Primitive characters and separable Gauss sums

8.11 The finite Fourier series of the Dirichlet characters

8.12 Polya's inequality for the partial sums of primitive characters

Chapter 9. Quadratic Residues and the Quadratic Reciprocity Law

9.1 Quadratic residues

9.2 Legendre's symbol and its properties

9.3 Evaluation of $\paren {-1 \mid p}$ and $\paren {2 \mid p}$

9.4 Gauss' lemma

9.5 The quadratic reciprocity law

9.6 Applications of the reciprocity law

9.7 The Jacobi symbol

9.8 Applications to Diophantine equations

9.9 Gauss sums and the quadratic reciprocity law

9.10 The reciprocity law for quadratic Gauss sums

9.11 Another proof of the quadratic reciprocity law

Chapter 10. Primitive Roots

10.1 The exponent of a number mod $m$. Primitive roots

10.2 Primitive roots and reduced residue systems

10.3 The nonexistence of primitive roots mod $2^\alpha$ for $\alpha \ge 3$

10.4 The existence of primitive roots mod $p$ for odd primes $p$

10.5 Primitive roots and quadratic residues

10.6 The existence of primitive roots mod $p^\alpha$

10.7 The existence of primitive roots mod $2 p^\alpha$

10.8 The nonexistence of primitive roots in the remaining cases

10.9 The number of primitive roots mod $m$

10.10 The index calculus

10.11 Primitive roots and Dirichlet characters

10.12 Real-valued Dirichlet characters mod $p^\alpha$

10.13 Primitive Dirichlet characters mod $p^\alpha$

Chapter 11. Dirichlet Series and Euler Products

11.1 Introduction

11.2 The half-plane of absolute convergence of a Dirichlet series

11.3 The function defined by a Dirichlet series

11.4 Multiplication of Dirichlet series

11.5 Euler products

11.6 The half-plane of convergence of a Dirichlet series

11.7 Analytic properties of Dirichlet series

11.8 Dirichlet series with nonnegative coefficients

11.9 Dirichlet series expressed as exponentials of Dirichlet series

11.10 Mean value formulas for Dirichlet series

11.11 An integral formula for the coefficients of a Dirichlet series

11.12 An integral formula for the partial sums of a Dirichlet series

Chapter 12. The Functions $\map \zeta s$ and $\map L {s, \chi}$

12.1 Introduction

12.2 Properties of the gamma function

12.3 Integral representation for the Hurwitz zeta function

12.4 A contour integral representation for the Hurwitz zeta function

12.5 The analytic continuation of the Hurwitz zeta function

12.6 Analytic continuation of $\map \zeta s$ and $\map L {s, \chi}$

12.7 Hurwitz's formula for $\map \zeta {s, a}$

12.8 The functional equation for the Riemann zeta function

12.9 A functional equation for the Hurwitz zeta function

12.10 The functional equation for $L$-functions

12.11 Evaluation of $\map \zeta {-n, a}$

12.12 Properties of Bernoulli numbers and Bernoulli polynomials

12.13 Formulas for $\map L {0, \chi}$

12.14 Approximation of $\map \zeta {s, a}$ by finite sums

12.15 Inequalities for $\size {\map \zeta {s, a} }$

12.16 Inequalities for $\size {\map \zeta s}$ and $\size {\map L {s, \chi} }$

Chapter 13. Analytic Proof of the Prime Number Theorem

13.1 The plan of the proof

13.2 Lemmas

13.3 A contour integral representation for $\map {\psi_1} x / x^2$

13.4 Upper bounds for $\size {\map \zeta s}$ and $\size {\map {\zeta'} s}$ near the line $\sigma = 1$

13.5 The nonvanishing of $\map \zeta s$ on the line $\sigma = 1$

13.6 Inequalities for $\size {1 / \map \zeta s}$ and $\size {\map {\zeta'} s / \map \zeta s}$

13.7 Completion of the proof of the prime number theorem

13.8 Zero-free regions for (?)

13.9 The Riemann hypothesis

13.10 Application to the divisor function

13.11 Application to Euler's totient

13.12 Extension of Polya's inequality for character sums

Chapter 14. Partitions

14.1 Introduction

14.2 Geometric representation of partitions

14.3 Generating functions for partitions

14.4 Euler's pentagonal-number theorem

14.5 Combinatorial proof of Euler's pentagonal-number theorem

14.6 Euler's recursion formula for $\map p n$

14.7 An upper bound for $\map p n$

14.8 Jacobi's triple product identity

14.9 Consequences of Jacobi's identity

14.10 Logarithmic differentiation of generating functions

14.11 The partition identities of Ramanujan

Bibliography

Index of Special Symbols

Index