This book has grown out of a course of lectures I have given at the Eidgenossische Technische Hochschule, Zurich. Notes of those lectures, prepared for the most part by assistants, have appeared in German. This book follows the same general plan as those notes, though in style, and in text (for instance, Chapters III, V, VIII), and in attention to detail, it is rather different. Its purpose is to introduce the non-specialist to some of the fundamental results in the theory of numbers, to show how analytical methods of proof fit into the theory, and to prepare the ground for a subsequent inquiry into deeper questions. It is pub­ lished in this series because of the interest evinced by Professor Beno Eckmann. I have to acknowledge my indebtedness to Professor Carl Ludwig Siegel, who has read the book, both in manuscript and in print, and made a number of valuable criticisms and suggestions. Professor Raghavan Narasimhan has helped me, time and again, with illuminating comments. Dr. Harold Diamond has read the proofs, and helped me to remove obscurities. I have to thank them all. K.C.

Inhalt

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 Möbius 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.