Indeholder "Preface", "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 μ(n)", "2.3 The Euler totient function φ(n)", "2.4 A relation connecting φ and μ", "2.5 A product formula for φ(n)", "2.6 The Dirichlet product of arithmetical functions", "2.7 Dirichlet inverses and the Mobius inversion formula", "2.8 The Mangoldt function Λ(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 λ(n)", "2.13 The divisor functions σ_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 σ_x(n)", "3.7 The average order of φ(n)", "3.8 An application to the distribution of lattice points visible from the origin", "3.9 The average order of μ(n) and of Λ(n)", "3.10 The partial sums of a Dirichlet product", "3.11 Applications to μ(n) and Λ(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 ψ(x) and Θ(x)", "4.3 Relations connecting Θ and π(x)", "4.4 Some equivalent forms of the prime number theorem", "4.5 Inequalities for π(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 Σ{p= 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^α", "10.7 The existence of primitive roots mod 2p^α", "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^α", "10.13 Primitive Dirichlet characters mod p^α", "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 ζ(s) and L(s,χ)", "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 ζ(s) and L(s,χ)", "12.7 Hurwitz's formula for ζ(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 ζ(-n,a)", "12.12 Properties of Bernoulli numbers and Bernoulli polynomials", "12.13 Formulas for L(0,χ)", "12.14 Approximation of ζ(s,a) by finite sums", "12.15 Inequalities for |ζ(s,a)|", "12.16 Inequalities for |ζ(s)| and |L(s,χ)|", "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 ψ_1(x)/x^2", "13.4 Upper bounds for |ζ(s)| and |ζ'(s)| near the line σ = 1", "13.5 The nonvanishing of ζ(s) on the line σ = 1", "13.6 Inequalities for |1/ζ(s)| and |ζ'(s)/ζ(s)|", "13.7 Completion of the proof of the prime number theorem", "13.8 Zero-free regions for ζ(s)", "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".

Denne bog var virkelig en hård nyser, da jeg fulgte et valgfrit modul om talteori, men den vinder ved nærmere bekendtskab. ( )