Introduction to analytic number theory

Includes index

Includes bibliographical references

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
7. Residue Classes
8. Theorems of Euler and of Fermat
9. The Number of Solutions of a Congruence
III. Rational Approximation of Irrationals and Hurwitz's Theorem
10. Approximation of Irrationals
11. Sums of Two Squares
12. Primes of the Form 4k±1
13. Hurwitz's Theorem
IV. Quadratic Residues and the Representation of a Number as a Sum of Four Squares
14. The Legendre Symbol
15. Wilson's Theorem and Euler's Criterion
16. Sums of Two Squares
17. Sums of Four Squares
V. The Law of Quadratic Reciprocity
18. Quadratic Reciprocity
19. Reciprocity for Generalized Gaussian Sums
20. Proof of Quadratic Reciprocity
21. Some Applications
VI. Arithmetical Functions and Lattice Points
22. Generalities
23. The Lattice Point Function r(n)
24. The Divisor Function d(n)
25. The Function τ(n)
26. The Möbius Function μ(n)
27. Euler's Function φ(n)
VII. Chebyshev's Theorem on the Distribution of Prime Numbers
28. The Chebyshev Functions
29. Chebyshev's Theorem
30. Bertrand's Postulate
31. Euler's Identity
32. Some Formulae of Mertens
VIII. Weyl's Theorems on Uniform Distribution and Kronecker's Theorem
33. Introduction
34. Uniform Distribution in the Unit Interval
35. Uniform Distribution Modulo 1
36. Weyl's Theorems
37. Kronecker's Theorem
IX. Minkowski's Theorem on Lattice Points in Convex Sets
38. Convex Sets
39. Minkowski's Theorem
40. Applications
X. Dirichlet's Theorem on Primes in an Arithmetical Progression
41. Introduction
42. Characters
43. Sums of Characters, Orthogonality Relations
44. Dirichlet Series, Landau's Theorem
45. Dirichlet's Theorem
XI. The Prime Number Theorem
46. The Non-Vanishing of ζ(1 + it)
47. The Wiener-Ikehara Theorem
48. The Prime Number Theorem

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.