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Preface | vii | |
Chapter 1 | Fundamental Concepts | |
1. | Divisibility | 1 |
2. | The gcd and the lcm | 4 |
3. | The Euclidean Algorithm | 7 |
4. | The Fundamental Theorem | 9 |
Chapter 2 | Arithmetic Functions | |
5. | The Semigroup A of Arithmetic Functions | 13 |
6. | The Group of Units in A | 16 |
7. | The Subgroup of Multiplicative Functions | 18 |
8. | The Mobius Function and Inversion Formulas | 20 |
9. | The Sigma Functions | 23 |
10. | The Euler [open phi]-Function | 27 |
Chapter 3 | Congruences and Residues | |
11. | Complete Residue Systems | 31 |
12. | Linear Congruences | 35 |
13. | Reduced Residue Systems | 39 |
14. | Ramanujan's Trigonometric Sum | 41 |
15. | Wilson's Theorem | 45 |
16. | Primitive Roots | 47 |
17. | Quadratic Residues | 52 |
18. | Congruences with Composite Moduli | 59 |
Chapter 4 | Summatory Functions | |
19. | Introduction | 63 |
20. | The Euler-McLaurin Sum Formula | 70 |
21. | Order of Magnitude of [tau](n) | 73 |
22. | Order of Magnitude of [sigma](n) | 76 |
23. | Sums Involving the Mobius Function | 78 |
24. | Squarefree Integers | 82 |
Chapter 5 | Sums of Squares | |
25. | Sums of Four Squares | 84 |
26. | Sums of Two Squares | 87 |
27. | Number of Representations | 90 |
28. | The Gaussian Integers | 90 |
29. | Proof of Theorem 27.1 | 95 |
30. | Restatement of Theorem 27.1 | 96 |
Chapter 6 | Continued Fractions, Farey Sequences, the Pell Equation | |
31. | Finite Continued Fractions | 98 |
32. | Infinite Simple Continued Fractions | 103 |
33. | Farey Sequences | 107 |
34. | The Pell Equation | 110 |
35. | Rational Approximations of Reals | 114 |
Chapter 7 | The Equation x[superscript n] + y[superscript n] = z[superscript n], n [less than or equal] 4 | |
36. | Pythagorean Triples | 120 |
37. | The Equation x[superscript 4] + y[superscript 4] = z[superscript 4] | 122 |
38. | Arithmetic in K([square root] -3) | 123 |
39. | The Equation x[superscript 3] + y[superscript 3] = z[superscript 3] | 126 |
Chapter 8 | The Prime Number Theorem | |
40. | Introductory Remarks | 129 |
41. | Preliminary Results | 130 |
42. | The Function [psi](x) | 134 |
43. | A Fundamental Inequality | 140 |
44. | The Behavior of r(x)/x | 144 |
45. | The Prime Number Theorem and Related Results | 153 |
Chapter 9 | Geometry of Numbers | |
46. | Preliminaries | 160 |
47. | Convex Symmetric Distance Functions | 164 |
48. | The Theorems of Minkowski | 170 |
49. | Applications to Farey Sequences and Continued Fractions | 174 |
Solutions for Selected Exercises | 181 | |
Bibliography | 203 | |
Index | 205 |
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Add The theory of numbers, Support text for a first course in number theory features the use of algebraic methods for studying arithmetic functions. Subjects covered include the Erdös-Selberg proof of the Prime Number Theorem, an introduction to algebraic and geometric number theor, The theory of numbers to the inventory that you are selling on WonderClubX
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Add The theory of numbers, Support text for a first course in number theory features the use of algebraic methods for studying arithmetic functions. Subjects covered include the Erdös-Selberg proof of the Prime Number Theorem, an introduction to algebraic and geometric number theor, The theory of numbers to your collection on WonderClub |