Introduction to Analytic Number Theory

I The unique factorization theorem.- 1. Primes.- 2. The unique factorization theorem.- 3. A second proof of Theorem 2.- 4. Greatest common divisor and least common multiple.- 5. Farey sequences.- 6. The infinitude of primes.- II Congruences.- 1. Residue classes.- 2. Theorems of Euler and of Fermat.- 3. The number of solutions of a congruence.- III Rational approximation of irrationals and Hurwitz's theorem.- 1. Approximation of irrationals.- 2. Sums of two squares.- 3. Primes of the form 4k+-.- 4. Hurwitz's theorem.- IV Quadratic residues and the representation of a number as a sum of four squares.- 1. The Legendre symbol.- 2. Wilson's theorem and Euler's criterion.- 3. Sums of two squares.- 4. Sums of four squares.- V The law of quadratic reciprocity.- 1. Quadratic reciprocity.- 2. Reciprocity for generalized Gaussian sums.- 3. Proof of quadratic reciprocity.- 4. Some applications.- VI Arithmetical functions and lattice points.- 1. Generalities.- 2. The lattice point function r(n).- 3. The divisor function d(n).- 4. The functions ?(n).- 5. The Mobius functions ?(n).- 6. Euler's function ?(n).- VII Chebyshev's therorem on the distribution of prime numbers.- 1. The Chebyshev functions.- 2. Chebyshev's theorem.- 3. Bertrand's postulate.- 4. Euler's identity.- 5. Some formulae of Mertens.- VIII Weyl's theorems on uniforms distribution and Kronecker's theorem.- 1. Introduction.- 2. Uniform distribution in the unit interval.- 3. Uniform distribution modulo 1.- 4. Weyl's theorems.- 5. Kronecker's theorem.- IX Minkowski's theorem on lattice points in convex sets.- 1. Convex sets.- 2. Minkowski's theorem.- 3. Applications.- X Dirichlet's theorem on primes in an arithmetical progression.- 1. Introduction.- 2. Characters.- 3. Sums of characters, orthogonality relations.- 4. Dirichlet series, Landau's theorem.- 5. Dirichlet's theorem.- XI The prime number theorem.- 1. The non-vanishing of ? (1 + it).- 2. The Wiener-Ikehara theorem.- 3. The prime number theorem.- A list of books.- Notes.