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

## Tom M. Apostol: *Introduction to Analytic Number Theory*

Published $1976$, **Springer-Verlag**.

### Subject Matter

### 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 $\mu(n)$
- 2.3 The Euler totient function $\phi(n)$
- 2.4 A relation connecting $\mu$ and $\phi$
- 2.5 A product formula for $\phi(n)$
- 2.6 The Dirichlet product of arithmetical functions
- 2.7 Dirichlet inverses and the Mobius inversion formula
- 2.8 The Mangoldt function $\Lambda(n)$
- 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 $\lambda(n)$
- 2.13 The divisor functions $\sigma_x(n)$
- 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 $d(n)$
- 3.6 The average order of the divisor functions $\sigma_x(n)$
- 3.7 The average order of $\phi(n)$
- 3.8 An application to the distribution of lattice points visible from the origin
- 3.9 The average order of $\mu(n)$ and of $\Lambda(n)$
- 3.10 The partial sums of a Dirichlet product
- 3.11 Applications to $\mu(n)$ and $\Lambda(n)$
- 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 $\psi(x)$ and $\vartheta(x)$
- 4.3 Relations connecting $\vartheta(x)$ and $\pi(x)$
- 4.4 Some equivalent forms of the prime number theorem
- 4.5 Inequalities for $\pi(n)$ 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 $\sum_{p\leq x}1/p$
- 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 $L(1,\chi)$ 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 $4n — 1$ and $4n + 1$
- 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 $s_k(n)$
- 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 $(-1 |p)$ and $(2|p)$
- 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 $\alpha \geq 3$
- 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 $2p^\alpha$
- 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 $\zeta(s)$ and $L(s,\chi)$
- 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 $\zeta(s)$ and $L(s,\chi)$
- 12.7 Hurwitz's formula for $\zeta(s,a)$
- 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 $\zeta(-n,a)$
- 12.12 Properties of Bernoulli numbers and Bernoulli polynomials
- 12.13 Formulas for $L(0, \chi)$
- 12.14 Approximation of $\zeta(s,a)$ by finite sums
- 12.15 Inequalities for $|\zeta(s,a)|$
- 12.16 Inequalities for $|\zeta(s)|$ and $|L(s,\chi)|$

- 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 $\psi_1(x)/x^2$
- 13.4 Upper bounds for $|\zeta(s)|$ and $|\zeta'(s)|$ near the line $\sigma = 1$
- 13.5 The nonvanishing of $\zeta(s)$ on the line $\sigma = 1$
- 13.6 Inequalities for $|1/\zeta(s)|$ and $|\zeta'(s)/\zeta(s)|$
- 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 $p(n)$
- 14.7 An upper bound for $p(n)$
- 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