Support text for a first course in number theory features the use of algebraic methods for studying arithmetic functions. Subjects covered include the Erd?s-Selberg proof of the Prime Number Theorem, an introduction to algebraic and geometric number theory—the former by studying Gaussian and Jacobian integers, the latter through geometric methods in proving the Quadratic Reciprocity Law and in proofs of certain asymptotic formulas for summatory functions.
Author(s): Anthony A. Gioia
Edition: Dover Ed
Year: 2001
Language: English
Pages: 226
Contents......Page 8
Preface......Page 5
1. Divisibility......Page 10
2. The gcd and the lcm......Page 13
3. The Euclidean Algorithm......Page 16
4. The Fundamental Theorem......Page 18
5. The Semigroup A of Arithmetic Functions......Page 22
6. The Group of Units in A......Page 25
7. The Subgroup of Multiplicative Functions......Page 27
8. The Möbius Function and Inversion Formulas......Page 29
9. The Sigma Functions......Page 32
10. The Euler \phi-Function......Page 36
11. Complete Residue Systems......Page 40
12. Linear Congruences......Page 44
13. Reduced Residue Systems......Page 48
14. Ramanujan's Trigonometric Sum......Page 50
15. Wilson's Theorem......Page 54
16. Primitive Roots......Page 56
17. Quadratic Residues......Page 61
18. Congruences with Composite Moduli......Page 68
19. Introduction......Page 72
20. The Euler-McLaurin Sum Formula......Page 79
21. Order of Magnitude of \tau(n)......Page 82
22. Order of Magnitude of \sigma(n)......Page 85
23. Sums Involving the Mobius Function......Page 87
24. Squarefree Integers......Page 91
25. Sums of Four Squares......Page 93
26. Sums of Two Squares......Page 96
28. The Gaussian Integers......Page 99
29. Proof of Theorem 27.1......Page 104
30. Restatement of Theorem 27.1......Page 105
31. Finite Continued Fractions......Page 107
32. Infinite Simple Continued Fractions......Page 112
33. Farey Sequences......Page 116
34. The Pell Equation......Page 119
35. Rational Approximations of Reals......Page 123
36. Pythagorean Triples......Page 129
37. The Equation x^4 + y^4 = z^4......Page 131
38. Arithmetic in K(\sqrt{-3})......Page 132
39. The Equation x^3 + y^3 = z^3......Page 135
40. Introductory Remarks......Page 138
41. Preliminary Results......Page 139
42. The Function \psi(x)......Page 143
43. A Fundamental Inequality......Page 149
44. The Behavior of r(x)/x......Page 153
45. The Prime Number Theorem and Related Results......Page 162
46. Preliminaries......Page 169
47. Convex Symmetric Distance Functions......Page 173
48. The Theorems of Minkowski......Page 179
49. Applications to Farey Sequences and Continued Fractions......Page 183
Solutions for Selected Exercises......Page 190
Bibliography......Page 211
Index......Page 212
