An Invitation to Modern Number Theory, Volume 13

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Princeton University Press, 2006 - Mathematics - 503 pages
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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.

 

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Contents

Mod p Arithmetic Group Theory and Cryptography
3
12 EFFICIENT ALGORITHMS
5
ARITHMETIC MODULO n
14
14 GROUP THEORY
15
15 RSA REVISITED
20
16 EISENSTEINS PROOF OF QUADRATIC RECIPROCITY
21
Arithmetic Functions
29
22 AVERAGE ORDER
32
114 APPLICATIONS OF THE FOURIER TRANSFORM
268
115 CENTRAL LIMIT THEOREM
273
116 ADVANCED TOPICS
276
nka and Poissonian Behavior
278
122 DENSENESS OF nka
280
123 EQUIDISTRIBUTION OF nka
283
124 SPACING PRELIMINARIES
288
125 POINT MASSES AND INDUCED PROBABILITY MEASURES
289

23 COUNTING THE NUMBER OF PRIMES
38
Zeta and LFunctions
47
32 ZEROS OF THE RIEMANN ZETA FUNCTION
54
33 DIRICHLET CHARACTERS AND LFUNCTIONS
69
Solutions to Diophantine Equations
81
42 ELLIPTIC CURVES
85
43 HEIGHT FUNCTIONS AND DIOPHANTINE EQUATIONS
89
44 COUNTING SOLUTIONS OF CONGRUENCES MODULO p
95
45 RESEARCH PROJECTS
105
Continued Fractions and Approximations
107
Algebraic and Transcendental Numbers
109
52 DEFINITIONS
110
53 COUNTABLE AND UNCOUNTABLE SETS
112
54 PROPERTIES OF e
118
55 EXPONENT OR ORDER OF APPROXIMATION
124
56 LIOUVILLES THEOREM
128
57 ROTHS THEOREM
132
The Proof of Roths Theorem
137
62 EQUIVALENT FORMULATION OF ROTHS THEOREM
138
63 ROTHS MAIN LEMMA
142
64 PRELIMINARIES TO PROVING ROTHS LEMMA
147
65 PROOF OF ROTHS LEMMA
155
Introduction to Continued Fractions
158
72 DEFINITION OF CONTINUED FRACTIONS
159
73 REPRESENTATION OF NUMBERS BY CONTINUED FRACTIONS
161
74 INFINITE CONTINUED FRACTIONS
167
75 POSITIVE SIMPLE CONVERGENTS AND CONVERGENCE
169
76 PERIODIC CONTINUED FRACTIONS AND QUADRATIC IRRATIONALS
170
77 COMPUTING ALGEBRAIC NUMBERS CONTINUED FRACTIONS
177
78 FAMOUS CONTINUED FRACTION EXPANSIONS
179
79 CONTINUED FRACTIONS AND APPROXIMATIONS
182
710 RESEARCH PROJECTS
186
Probabilistic Methods and Equidistribution
189
Introduction to Probability
191
81 PROBABILITIES OF DISCRETE EVENTS
192
82 STANDARD DISTRIBUTIONS
205
83 RANDOM SAMPLING
211
84 THE CENTRAL LIMIT THEOREM
213
Applications of Probability Benfords Law and Hypothesis Testing
216
92 BENFORDS LAW AND EQUIDISTRIBUTED SEQUENCES
218
93 RECURRENCE RELATIONS AND BENFORDS LAW
219
94 RANDOM WALKS AND BENFORDS LAW
221
95 STATISTICAL INFERENCE
225
96 SUMMARY
229
Distribution of Digits of Continued Fractions
231
102 MEASURE OF a WITH SPECIFIED DIGITS
235
103 THE GAUSSKUZMIN THEOREM
237
104 DEPENDENCIES OF DIGITS
244
105 GAUSSKUZMIN EXPERIMENTS
248
106 RESEARCH PROJECTS
252
Introduction to Fourier Analysis
255
111 INNER PRODUCT OF FUNCTIONS
256
112 FOURIER SERIES
258
113 CONVERGENCE OF FOURIER SERIES
262
126 NEIGHBOR SPACINGS
290
127 POISSONIAN BEHAVIOR
291
128 NEIGHBOR SPACINGS OF nka
296
129 RESEARCH PROJECTS
299
The Circle Method
301
Introduction to the Circle Method
303
132 THE CIRCLE METHOD
309
133 GOLDBACHS CONJECTURE REVISITED
315
Circle Method Heuristics for Germain Primes
326
142 PRELIMINARIES
328
143 THE FUNCTIONS FNx AND ux
331
144 APPROXIMATING FNx ON THE MAJOR ARCS
332
145 INTEGRALS OVER THE MAJOR ARCS
338
146 MAJOR ARCS AND THE SINGULAR SERIES
342
147 NUMBER OF GERMAIN PRIMES AND WEIGHTED SUMS
350
148 EXERCISES
353
149 RESEARCH PROJECTS
354
Random Matrix Theory and LFunctions
357
From Nuclear Physics to LFunctions
359
152 EIGENVALUE PRELIMINARIES
364
153 SEMICIRCLE LAW
368
154 ADJACENT NEIGHBOR SPACINGS
374
155 THIN SUBFAMILIES
377
156 NUMBER THEORY
383
157 SIMILARITIES BETWEEN RANDOM MATRIX THEORY AND L FUNCTIONS
389
158 SUGGESTIONS FOR FURTHER READING
390
Random Matrix Theory Eigenvalue Densities
391
162 NONSEMICIRCLE BEHAVIOR
398
163 SPARSE MATRICES
402
164 RESEARCH PROJECTS
403
Random Matrix Theory Spacings between Adjacent Eigenvalues
405
172 DISTRIBUTION OF EIGENVALUES OF 2 x 2 GOE MODEL
409
173 GENERALIZATION TO N x N GOE
414
174 CONJECTURES AND RESEARCH PROJECTS
418
The Explicit Formula and Density Conjectures
421
181 EXPLICIT FORMULA
422
182 DIRICHLET CHARACTERS FROM A PRIME CONDUCTOR
429
183 SUMMARY OF CALCULATIONS
437
Analysis Review
439
A2 CALCULUS REVIEW
442
A3 CONVERGENCE AND CONTINUITY
447
A4 DIRICHLETS PIGEONHOLE PRINCIPLE
448
A5 MEASURES AND LENGTH
450
A6 INEQUALITIES
452
Linear Algebra Review
455
B2 CHANGE OF BASIS
456
B3 ORTHOGONAL AND UNITARY MATRICES
457
B4 TRACE
458
B5 SPECTRAL THEOREM FOR REAL SYMMETRIC MATRICES
459
Hints and Remarks on the Exercises
463
Concluding Remarks
475
Bibliography
476
Index
497
Copyright

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Page 477 - PJ and KA Doksum Mathematical Statistics: Basic Ideas and Selected Topics. Holden-day, San Francisco (1977). 2. Melendez, M. "Detection of Gross Errors in Real Chemical Process Data.
Page 477 - The energy level spacing for two harmonic oscillators with generic ratio of frequencies, Journ.

About the author (2006)

Steven J. Miller is an Assistant Professor of Mathematics at Brown University. Ramin Takloo-Bighash is an Assistant Professor of Mathematics at Princeton University.

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