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An Invitation to Modern Number Theory : International Studen - Steven J. Miller

An Invitation to Modern Number Theory

International Studen

Hardcover Published: 26th March 2006
ISBN: 9780691120607
Number Of Pages: 528

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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.

Industry Reviews

"This is a great book... [I]t is a fine book for talented and mathematically mature undergraduates, for graduate students, and for anyone looking for information on modern number theory."--Henry Ricardo, MAA Reviews "This is the first text to present Random Matrix Theory and the Circle Method for German primes. This well-written book supplements classic texts by showing connections between seemingly diverse topics, by making the subject accessible to beginning students and by whetting their appetite for continuing in mathematics"--Mathematical Reviews "I would highly recommend this book to anybody interested in number theory, from an undergraduate student to an established expert, since everybody will be able to find in this book lots of new interesting material, tempting problems, and interesting computational challenges. It could also be used as a textbook for a graduate course in number theory. To promote and stimulate independent research, it contains many very interesting exercises and even suggestions for research projects."--Igor Shparlinski, SIAM Review

Forewordp. xi
Prefacep. xiii
Notationp. xix
Basic Number Theoryp. 1
Mod p Arithmetic, Group Theory and Cryptographyp. 3
Cryptographyp. 3
Efficient Algorithmsp. 5
Clock Arithmetic: Arithmetic Modulo np. 14
Group Theoryp. 15
RSA Revisitedp. 20
Eisenstein's Proof of Quadratic Reciprocityp. 21
Arithmetic Functionsp. 29
Arithmetic Functionsp. 29
Average Orderp. 32
Counting the Number of Primesp. 38
Zeta and L-Functionsp. 47
The Riemann Zeta Functionp. 47
Zeros of the Riemann Zeta Functionp. 54
Dirichlet Characters and L-Functionsp. 69
Solutions to Diophantine Equationsp. 81
Diophantine Equationsp. 81
Elliptic Curvesp. 85
Height Functions and Diophantine Equationsp. 89
Counting Solutions of Congruences Modulo pp. 95
Research Projectsp. 105
Continued Fractions and Approximationsp. 107
Algebraic and Transcendental Numbersp. 109
Russell's Paradox and the Banach-Tarski Paradoxp. 109
Definitionsp. 110
Countable and Uncountable Setsp. 112
Properties of ep. 118
Exponent (or Order) of Approximationp. 124
Liouville's Theoremp. 128
Roth's Theoremp. 132
The Proof of Roth's Theoremp. 137
Liouville's Theorem and Roth's Theoremp. 137
Equivalent Formulation of Roth's Theoremp. 138
Roth's Main Lemmap. 142
Preliminaries to Proving Roth's Lemmap. 147
Proof of Roth's Lemmap. 155
Introduction to Continued Fractionsp. 158
Decimal Expansionsp. 158
Definition of Continued Fractionsp. 159
Representation of Numbers by Continued Fractionsp. 161
Infinite Continued Fractionsp. 167
Positive Simple Convergents and Convergencep. 169
Periodic Continued Fractions and Quadratic Irrationalsp. 170
Computing Algebraic Numbers' Continued Fractionsp. 177
Famous Continued Fraction Expansionsp. 179
Continued Fractions and Approximationsp. 182
Research Projectsp. 186
Probabilistic Methods and Equidistributionp. 189
Introduction to Probabilityp. 191
Probabilities of Discrete Eventsp. 192
Standard Distributionsp. 205
Random Samplingp. 211
The Central Limit Theoremp. 213
Applications of Probability: Benford's Law and Hypothesis Testingp. 216
Benford's Lawp. 216
Benford's Law and Equidistributed Sequencesp. 218
Recurrence Relations and Benford's Lawp. 219
Random Walks and Benford's Lawp. 221
Statistical Inferencep. 225
Summaryp. 229
Distribution of Digits of Continued Fractionsp. 231
Simple Results on Distribution of Digitsp. 231
Measure of [alpha] with Specified Digitsp. 235
The Gauss-Kuzmin Theoremp. 237
Dependencies of Digitsp. 244
Gauss-Kuzmin Experimentsp. 248
Research Projectsp. 252
Introduction to Fourier Analysisp. 255
Inner Product of Functionsp. 256
Fourier Seriesp. 258
Convergence of Fourier Seriesp. 262
Applications of the Fourier Transformp. 268
Central Limit Theoremp. 273
Advanced Topicsp. 276
{n[superscript k alpha]} and Poissonian Behaviorp. 278
Definitions and Problemsp. 278
Denseness of {n[superscript k alpha]}p. 280
Equidistribution of {n[superscript k alpha]}p. 283
Spacing Preliminariesp. 288
Point Masses and Induced Probability Measuresp. 289
Neighbor Spacingsp. 290
Poissonian Behaviorp. 291
Neighbor Spacings of {n[superscript k alpha]}p. 296
Research Projectsp. 299
The Circle Methodp. 301
Introduction to the Circle Methodp. 303
Originsp. 303
The Circle Methodp. 309
Goldbach's Conjecture Revisitedp. 315
Circle Method: Heuristics for Germain Primesp. 326
Germain Primesp. 326
Preliminariesp. 328
The Functions F[subscript N](x) and u(x)p. 331
Approximating F[subscript N](x) on the Major Arcsp. 332
Integrals over the Major Arcsp. 338
Major Arcs and the Singular Seriesp. 342
Number of Germain Primes and Weighted Sumsp. 350
Exercisesp. 353
Research Projectsp. 354
Random Matrix Theory and L-Functionsp. 357
From Nuclear Physics to L-Functionsp. 359
Historical Introductionp. 359
Eigenvalue Preliminariesp. 364
Semi-Circle Lawp. 368
Adjacent Neighbor Spacingsp. 374
Thin Sub-familiesp. 377
Number Theoryp. 383
Similarities between Random Matrix Theory and L-Functionsp. 389
Suggestions for Further Readingp. 390
Random Matrix Theory: Eigenvalue Densitiesp. 391
Semi-Circle Lawp. 391
Non-Semi-Circle Behaviorp. 398
Sparse Matricesp. 402
Research Projectsp. 403
Random Matrix Theory: Spacings between Adjacent Eigenvaluesp. 405
Introduction to the 2 x 2 GOE Modelp. 405
Distribution of Eigenvalues of 2 x 2 GOE Modelp. 409
Generalization to N x N GOEp. 414
Conjectures and Research Projectsp. 418
The Explicit Formula and Density Conjecturesp. 421
Explicit Formulap. 422
Dirichlet Characters from a Prime Conductorp. 429
Summary of Calculationsp. 437
Analysis Reviewp. 439
Proofs by Inductionp. 439
Calculus Reviewp. 442
Convergence and Continuityp. 447
Dirichlet's Pigeon-Hole Principlep. 448
Measures and Lengthp. 450
Inequalitiesp. 452
Linear Algebra Reviewp. 455
Definitionsp. 455
Change of Basisp. 456
Orthogonal and Unitary Matricesp. 457
Tracep. 458
Spectral Theorem for Real Symmetric Matricesp. 459
Hints and Remarks on the Exercisesp. 463
Concluding Remarksp. 475
Bibliographyp. 476
Indexp. 497
Table of Contents provided by Ingram. All Rights Reserved.

ISBN: 9780691120607
ISBN-10: 0691120609
Series: International Studen
Audience: Tertiary; University or College
Format: Hardcover
Language: English
Number Of Pages: 528
Published: 26th March 2006
Publisher: Princeton University Press
Country of Publication: US
Dimensions (cm): 23.98 x 16.1  x 3.96
Weight (kg): 0.85

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