An Invitation to Modern Number Theory (Hardcover)

Steven J. Miller, Ramin Takloo-Bighash

  • 出版商: Princeton University
  • 出版日期: 2006-03-26
  • 售價: $1,280
  • 貴賓價: 9.8$1,254
  • 語言: 英文
  • 頁數: 519
  • 裝訂: Hardcover
  • ISBN: 0691120609
  • ISBN-13: 9780691120607
  • 下單後立即進貨 (約5~7天)

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Description

In a manner accessible to beginning undergraduates, An Invitation to Modern Number Theory introduces many of the central problems, conjectures, results, and techniques of the field, such as the Riemann Hypothesis, Roth's Theorem, the Circle Method, and Random Matrix Theory. Showing how experiments are used to test conjectures and prove theorems, the book allows students to do original work on such problems, often using little more than calculus (though there are numerous remarks for those with deeper backgrounds). It shows students what number theory theorems are used for and what led to them and suggests problems for further research.

Steven Miller and Ramin Takloo-Bighash introduce the problems and the computational skills required to numerically investigate them, providing background material (from probability to statistics to Fourier analysis) whenever necessary. They guide students through a variety of problems, ranging from basic number theory, cryptography, and Goldbach's Problem, to the algebraic structures of numbers and continued fractions, showing connections between these subjects and encouraging students to study them further. In addition, this is the first undergraduate book to explore Random Matrix Theory, which has recently become a powerful tool for predicting answers in number theory.

Providing exercises, references to the background literature, and Web links to previous student research projects, An Invitation to Modern Number Theory can be used to teach a research seminar or a lecture class.

 

Table Of Contents

 

Foreword xi
Preface xiii
Notation xix

PART 1. BASIC NUMBER THEORY 1

Chapter 1. Mod p Arithmetic, Group Theory and Cryptography 3
Chapter 2. Arithmetic Functions 29
Chapter 3. Zeta and L-Functions 47
Chapter 4. Solutions to Diophantine Equations 81

PART 2. CONTINUED FRACTIONS AND APPROXIMATIONS 107

Chapter 5. Algebraic and Transcendental Numbers 109
Chapter 6. The Proof of Roth's Theorem 137
Chapter 7. Introduction to Continued Fractions 158

PART 3. PROBABILISTIC METHODS AND EQUIDISTRIBUTION 189

Chapter 8. Introduction to Probability 191
Chapter 9. Applications of Probability: Benford's Law and Hypothesis Testing 216
Chapter 10. Distribution of Digits of Continued Fractions 231
Chapter 11. Introduction to Fourier Analysis 255
Chapter 12. f n k g and Poissonian Behavior 278

PART 4. THE CIRCLE METHOD 301

Chapter 13. Introduction to the Circle Method 303
Chapter 14. Circle Method: Heuristics for Germain Primes 326

PART 5. RANDOM MATRIX THEORY AND L-FUNCTIONS 357

Chapter 15. From Nuclear Physics to L-Functions 359
Chapter 16. Random Matrix Theory: Eigenvalue Densities 391
Chapter 17. Random Matrix Theory: Spacings between Adjacent Eigenvalues 405
Chapter 18. The Explicit Formula and Density Conjectures 421

Appendix A. Analysis Review 439
Appendix B. Linear Algebra Review 455
Appendix C. Hints and Remarks on the Exercises 463
Appendix D. Concluding Remarks 475

Bibliography 476
Index 497

商品描述(中文翻譯)

《現代數論邀請函》以淺顯易懂的方式介紹了該領域的許多核心問題、猜想、結果和技巧,如黎曼猜想、羅斯定理、圓方法和隨機矩陣理論。本書展示了如何使用實驗來測試猜想和證明定理,讓學生能夠在這些問題上進行原創性的工作,通常只需要使用微積分(雖然對於有更深入背景的學生,也有許多備註)。它向學生展示了數論定理的應用和發展,並提出了進一步研究的問題。

史蒂文·米勒(Steven Miller)和拉敏·塔克盧-比加什(Ramin Takloo-Bighash)介紹了解決這些問題所需的計算技巧,並在必要時提供背景材料(從概率到統計到傅立葉分析)。他們引導學生解決各種問題,從基礎數論、密碼學和哥德巴赫猜想,到數字的代數結構和連分數,展示了這些主題之間的聯繫,並鼓勵學生進一步研究。此外,這是第一本探索隨機矩陣理論的本科生教材,該理論最近成為預測數論答案的強大工具。

《現代數論邀請函》提供練習題、背景文獻的參考和以前學生研究項目的網絡鏈接,可用於教授研究研討會或講座課程。

目錄:

前言
序言
符號說明

第一部分:基礎數論
第1章:模p算術、群論和密碼學
第2章:算術函數
第3章:黎曼ζ函數和L-函數
第4章:丟番圖方程的解

第二部分:連分數和逼近
第5章:代數和超越數
第6章:羅斯定理的證明
第7章:連分數的介紹

第三部分:概率方法和均勻分佈
第8章:概率的介紹
第9章:概率的應用:本福德定律和假設檢驗
第10章:連分數的數字分佈
第11章:傅立葉分析的介紹
第12章:f_n(k)和泊松行為

第四部分:圓方法
第13章:圓方法的介紹
第14章:圓方法:對於傑曼質數的啟發式方法

第五部分:隨機矩陣理論和L-函數
第15章:從核物理到L-函數
第16章:隨機矩陣理論:特徵值密度
第17章:隨機矩陣理論:相鄰特徵值之間的間距
第18章:```