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

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

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

### 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
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
9.6 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