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

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