|
|
Sold Out
Book Categories |
| 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 |
| Price | Condition | Delivery | Seller | Action |
| $99.99 | Digital |
| WonderClub |
Jan Michael Magbag
reviewed The theory of numbers on January 28, 2012
The theory of numbers
The Riemann Hypothesis is easily one of the most important unsolved problems in mathematics. Unfortunately, unlike Fermat's Last Theorem or even the Poincaré conjecture, the Riemann Hypothesis doesn't lend itself well to being put in layman's terms. Even still, Sabbagh gives quite the valiant effort in his book, and in the process takes us into the strange world of cutting-edge mathematics.
Sabbagh's trip through the Riemann Hypothesis is a slow, methodical one: he starts out with some required fundamentals: prime numbers, imaginary numbers, complex numbers, and works his way up to more esoteric topics like infinite series, zeta functions, and finally what Riemann's Hypothesis says about the zeros of these functions. Along the way, Sabbagh introduces various mathematicians whom, although you've probably never heard them, are larger-than-life in their own right, and what life is like for these people who concentrate intensely on very abstract concepts.
As I've mentioned in other reviews, I love books that talk about the 'romantic' side of mathematics, and Sabbagh does a wonderful job of that in his book. Many of the mathematicians he interviewed I had never heard of before, but their stories were so enthralling that I never once felt a twinge of boredom.
My only real criticisms of the book lie in it's flow. I know that trying to explain the Riemann Hypothesis in non-mathematical language is a very daunting task, but the explanations and analogies scattered throughout the book didn't seem to flow well enough to paint an overall picture for me. Also, in some areas, Sabbagh seemed to diverge quite a bit from the main topic of the book. Although the segues themselves are interesting topics, again I felt it disrupted the overall flow.
If you're like me, and you followed the stories behind Fermat's Last Theorem and the Poincare conjecture, I definitely recommend picking up Sabbagh's book and discovering the weird and fascinating world of Riemann's Hypothesis.
Login|Complaints|Blog|Games|Digital Media|Souls|Obituary|Contact Us|FAQ
CAN'T FIND WHAT YOU'RE LOOKING FOR? CLICK HERE!!! X
You must be logged in to add to WishlistX
This item is in your Wish List
X
This item is in your Collection
The theory of numbers
X
This Item is in Your Inventory
The theory of numbers
X
You must be logged in to review the productsX
X
X
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
X
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 |