Elementary methods in number theory:
Saved in:
| Main Author: | |
|---|---|
| Format: | Book |
| Language: | English |
| Published: |
New York [u.a.]
Springer
2000
|
| Series: | Graduate texts in mathematics
195 |
| Subjects: | |
| Online Access: | Inhaltsverzeichnis |
| Item Description: | Literaturverz. S. 497 - 507 |
| Physical Description: | XVIII, 513 S. |
| ISBN: | 0387989129 |
Staff View
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Record in the Search Index
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| adam_text | Contents
Preface vii
Notation and conventions xi
I A First Course in Number Theory 1
1 Divisibility and Primes 3
1.1 Division Algorithm 3
1.2 Greatest Common Divisors 10
1.3 The Euclidean Algorithm and Continued Fractions 17
1.4 The Fundamental Theorem of Arithmetic 25
1.5 Euclid s Theorem and the Sieve of Eratosthenes 33
1.6 A Linear Diophantine Equation 37
1.7 Notes 42
2 Congruences 45
2.1 The Ring of Congruence Classes 45
2.2 Linear Congruences 51
2.3 The Euler Phi Function 57
2.4 Chinese Remainder Theorem 61
2.5 Euler s Theorem and Fermat s Theorem 67
2.6 Pseudoprimes and Carmichael Numbers 74
2.7 Public Key Cryptography 76
xvi Contents
2.8 Notes 80
3 Primitive Roots and Quadratic Reciprocity 83
3.1 Polynomials and Primitive Roots 83
3.2 Primitive Roots to Composite Moduli 91
3.3 Power Residues 98
3.4 Quadratic Residues 100
3.5 Quadratic Reciprocity Law 109
3.6 Quadratic Residues to Composite Moduli 116
3.7 Notes 120
4 Fourier Analysis on Finite Abelian Groups 121
4.1 The Structure of Finite Abelian Groups 121
4.2 Characters of Finite Abelian Groups 126
4.3 Elementary Fourier Analysis 133
4.4 Poisson Summation 140
4.5 Trace Formulae on Finite Abelian Groups 144
4.6 Gauss Sums and Quadratic Reciprocity 151
4.7 The Sign of the Gauss Sum 160
4.8 Notes 169
5 The abc Conjecture 171
5.1 Ideals and Radicals 171
5.2 Derivations 175
5.3 Mason s Theorem 181
5.4 The abc Conjecture 185
5.5 The Congruence abc Conjecture 191
5.6 Notes 196
II Divisors and Primes in Multiplicative Number
Theory 199
6 Arithmetic Functions 201
6.1 The Ring of Arithmetic Functions 201
6.2 Mean Values of Arithmetic Functions 206
6.3 The Mobius Function 217
6.4 Multiplicative Functions 224
6.5 The mean value of the Euler Phi Function 227
6.6 Notes 229
7 Divisor Functions 231
7.1 Divisors and Factorizations 231
7.2 A Theorem of Ramanujan 237
7.3 Sums of Divisors 240
Contents xvii
7.4 Sums and Differences of Products 246
7.5 Sets of Multiples 255
7.6 Abundant Numbers 260
7.7 Notes 265
8 Prime Numbers 267
8.1 Chebyshev s Theorems 267
8.2 Mertens s Theorems 275
8.3 The Number of Prime Divisors of an Integer 282
8.4 Notes 287
9 The Prime Number Theorem 289
9.1 Generalized Von Mangoldt Functions 289
9.2 Selberg s Formulae 293
9.3 The Elementary Proof 299
9.4 Integers with k Prime Factors 313
9.5 Notes 320
10 Primes in Arithmetic Progressions 325
10.1 Dirichlet Characters 325
10.2 Dirichlet L Functions 330
10.3 Primes Modulo 4 338
10.4 The Nonvanishing of 1(1, x) 341
10.5 Notes 350
III Three Problems in Additive Number Theory 353
11 Waring s Problem 355
11.1 Sums of Powers 355
11.2 Stable Bases 359
11.3 Shnirel man s Theorem 361
11.4 Waring s Problem for Polynomials 367
11.5 Notes 373
12 Sums of Sequences of Polynomials 375
12.1 Sums and Differences of Weighted Sets 375
12.2 Linear and Quadratic Equations 382
12.3 An Upper Bound for Representations 387
12.4 Waring s Problem for Sequences of Polynomials 394
12.5 Notes 398
13 Liouville s Identity 401
13.1 A Miraculous Formula 401
13.2 Prime Numbers and Quadratic Forms 404
13.3 A Ternary Form 411
xviii Contents
13.4 Proof of Liouville s Identity 413
13.5 Two Corollaries 419
13.6 Notes 421
14 Sums of an Even Number of Squares 423
14.1 Summary of Results 423
14.2 A Recursion Formula 424
14.3 Sums of Two Squares 427
14.4 Sums of Four Squares 431
14.5 Sums of Six Squares 436
14.6 Sums of Eight Squares 441
14.7 Sums of Ten Squares 445
14.8 Notes 453
15 Partition Asymptotics 455
15.1 The Size of p(n) 455
15.2 Partition Functions for Finite Sets 458
15.3 Upper and Lower Bounds for logp(n) 465
15.4 Notes 473
16 An Inverse Theorem for Partitions 475
16.1 Density Determines Asymptotics 475
16.2 Asymptotics Determine Density 482
16.3 Abelian and Tauberian Theorems 486
16.4 Notes 495
References 497
Index 509
|
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| indexdate | 2024-07-09T18:38:44Z |
| institution | BVB |
| isbn | 0387989129 |
| language | English |
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| physical | XVIII, 513 S. |
| publishDate | 2000 |
| publishDateSearch | 2000 |
| publishDateSort | 2000 |
| publisher | Springer |
| record_format | marc |
| series | Graduate texts in mathematics |
| series2 | Graduate texts in mathematics |
| spelling | Nathanson, Melvyn B. Verfasser aut Elementary methods in number theory Melvyn B. Nathanson New York [u.a.] Springer 2000 XVIII, 513 S. txt rdacontent n rdamedia nc rdacarrier Graduate texts in mathematics 195 Literaturverz. S. 497 - 507 Zahlentheorie Number theory Zahlentheorie (DE-588)4067277-3 gnd rswk-swf Zahlentheorie (DE-588)4067277-3 s DE-604 Graduate texts in mathematics 195 (DE-604)BV000000067 195 HBZ Datenaustausch application/pdf http://bvbr.bib-bvb.de:8991/F?func=service&doc_library=BVB01&local_base=BVB01&doc_number=008910977&sequence=000002&line_number=0001&func_code=DB_RECORDS&service_type=MEDIA Inhaltsverzeichnis |
| spellingShingle | Nathanson, Melvyn B. Elementary methods in number theory Graduate texts in mathematics Zahlentheorie Number theory Zahlentheorie (DE-588)4067277-3 gnd |
| subject_GND | (DE-588)4067277-3 |
| title | Elementary methods in number theory |
| title_auth | Elementary methods in number theory |
| title_exact_search | Elementary methods in number theory |
| title_full | Elementary methods in number theory Melvyn B. Nathanson |
| title_fullStr | Elementary methods in number theory Melvyn B. Nathanson |
| title_full_unstemmed | Elementary methods in number theory Melvyn B. Nathanson |
| title_short | Elementary methods in number theory |
| title_sort | elementary methods in number theory |
| topic | Zahlentheorie Number theory Zahlentheorie (DE-588)4067277-3 gnd |
| topic_facet | Zahlentheorie Number theory |
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