Historical IntroductionChapter 1The Fundamental Theorem of Arithmetic1.1 Introduction1.2 Divisibility1.3 Greatest common divisor1.4 Prime numbers1.5 The fundamental theorem of arithmetic1.6 The series of reciprocals of the primes1.7 The Euclidean algorithm1.8 The greatest common divisor of more than two, numbers Exercises for Chapter 1Chapter 2Arithmetical Functions and Dirichlet Multiplication2.1 Introduction2.2 The M6bius function (n)2.3 The Euler totient function (n)2.4 A relation connecting and u2.5 A product formula for (n)2.6 The Dirichlet product of arithmetical functions2.7 Dirichlet inverses and the M6bius inversion formula2.8 The Mangoldt function A(n)2.9 Muitiplicative functions2.10 Multiplicative functions and Dirichlet multiplication2.11 The inverse of a completely multiplicative function2.12 Liouville's function)2.13 The divisor functions a,(n)2.14 Generalized convolutions2.15 Formal power series2.16 The Bell series of an arithmetical function2.17 Bell series and Dirichlet multiplication2.18 Derivatives of arithmetical functions2.19 The Selberg identity Exercises for Chapter 2Chapter 3Averages of Arithmetical Functions3.1 Introduction3.2 The big oh notation. Asymptotic equality of functions3.3 Euler's summation formula3.4 Some elementary asymptotic formulas3.5 The average order of din)3.6 The average order of the divisor functions a,(n)3.7 The average order of ~0(n)3.8 An application to the distribution of lattice points visible from the origin3.9 The average order of/4n) and of A(n)3.10 The partial sums ofa Dirichlet product3.11 Applications to pin) and A(n)3.12 Another identity for the partial gums of a Dirichlet product Exercises for Chapter 3Chapter 4Some Elementary Theorems on the Distribution of PrimeNumbers4.1 Introduction4.2 Chebyshev's functions (x) and (x)4.3 Relations connecting/x) and n(x)4.4 Some equivalent forms of the prime number theorem4.5 Inequalities for (n) and p,4.6 Shapiro's Tauberian theorem4.7 Applications of Shapiro's theorem4.8 An asymptotic formula for the partial sums, (I/p)4.9 The partial sums of the M6bius function 914.10 Brief sketch of an elementary proof of the prime number theorem4.11 Selbcrg's asymptotic formula Exercises for Chapter 4Chapter 5Congruences 5.1 Definition and basic properties of congruences 5.2 Residue classes and complete residue systems 5.3 Linear congruencesChapter 6Finite Abelian Groups and Their CharactersChapter 7Dirichlet's Theorem on Primes in Arithmetic ProgressionsChapter 8Periodic Arithmetical Functions and Gauss SumsChapter 9Quadratic Residues and the Quadratic Reciprocity LawChapter 10Primitive RootsChapter 11Dirichlet Series and Euler ProductsChapter 12The Functions (s) and L(s,x)Chapter 13Analytic Proof of the Prime Number Theorem