哈代数论 英文版PDF电子书下载

Ⅰ. THE SERIES OF PRIMES（1） 1

1.1. Divisibility of integers 1

1.2. Prime numbers 2

1.3. Statement of the fundamental theorem of arithmetic 3

1.4. The sequence of primes 4

1.5. Some questions concerning primes 6

1.6. Some notations 7

1.7. The logarithmic function 9

1.8. Statement of the prime number theorem 10

Ⅱ. THE SERIES OF PRIMES（2） 14

2.1. First proof of Euclid，s second theorem 14

2.2. Further deductions from Euclid，s argument 14

2.3. Primes in certain arithmetical progressions 15

2.4. Second proof of Euclid，s theorem 17

2.5. Fermat，s and Mersenne，s numbers 18

2.6. Third proof of Euclid，s theorem 20

2.7. Further results on formulae for primes 21

2.8. Unsolved problems concerning primes 23

2.9. Moduli of integers 23

2.10. Proof of the fundamental theorem of arithmetic 25

2.11. Another proof of the fundamental theorem 26

Ⅲ. FAREY SERIES AND A THEOREM OF MINKOWSKI 28

3.1. The definition and simplest properties of a Farey series 28

3.2. The equivalence of the two characteristic properties 29

3.3. First proof of Theorems 28 and 29 30

3.4. Second proof of the theorems 31

3.5. The integral lattice 32

3.6. Some simple properties of the fundamental lattice 33

3.7. Third proof of Theorems 28 and 29 35

3.8. The Farey dissection of the continuum 36

3.9. A theorem of Minkowski 37

3.10. Proof of Minkowski，s theorem 39

3.11. Developments of Theorem 37 40

Ⅳ. IRRATIONAL NUMBERS 45

4.1. Some generalities 45

4.2. Numbers known to be irrational 46

4.3. The theorem of Pythagoras and its generalizations 47

4.4. The use of the fundamental theorem in the proofs of Theorems 43-45 49

4.5. A historical digression 50

4.6. Geometrical proof of the irrationality of √5 52

4.7. Some more irrational numbers 53

Ⅴ. CONGRUENCES AND RESIDUES 57

5.1. Highest common divisor and least common multiple 57

5.2. Congruences and classes of residues 58

5.3. Elementary properties of congruences 60

5.4. Linear congruences 60

5.5. Euler，s function φ（m） 63

5.6. Applications of Theorems 59 and 61 to trigonometrical sums 65

5.7. A general principle 70

5.8. Construction of the regular polygon of 17 sides 71

Ⅵ. FERMAT，S THEOREM AND ITS CONSEQUENCES 78

6.1. Fermat，s theorem 78

6.2. Some properties of binomial coeffcients 79

6.3. A second proof of Theorem 72 81

6.4. Proof of Theorem 22 82

6.5. Quadratic residues 83

6.6. Special cases of Theorem 79： Wilson，s theorem 85

6.7. Elementary properties of quadratic residues and non-residues 87

6.8. The order of a （mod m） 88

6.9. The converse of Fermat，s theorem 89

6.10. Divisibility of 2p-1 -1 by p2 91

6.11. Gauss，s lemma and the quadratic character of 2 92

6.12. The law of reciprocity 95

6.13. Proof of the law of reciprocity 97

6.14. Tests for primality 98

6.15. Factors of Mersenne numbers； a theorem of Euler 100

Ⅶ. GENERAL PROPERTIES OF CONGRUENCES 103

7.1. Roots of congruences 103

7.2. Integral polynomials and identical congruences 103

7.3. Divisibility of polynomials （mod m） 105

7.4. Roots of congruences to a prime modulus 106

7.5. Some applications of the general theorems 108

7.6. Lagrange，s proof of Fermat，s and Wilson，s theorems 110

7.7. The residue of ｛1/2（p-1）｝！ 111

7.8. A theorem of Wolstenholme 112

7.9. The theorem of von Staudt 115

7.10. Proof of von Staudt，s theorem 116

Ⅷ. CONGRUENCES TO COMPOSITE MODULI 120

8.1. Linear congruences 120

8.2. Congruences of higher degree 122

8.3. Congruences to a prime-power modulus 123

8.4. Examples 125

8.5. Bauer，s identical congruence 126

8.6. Bauer，s congruence： the case p=2 129

8.7. A theorem of Leudesdorf 130

8.8. Further consequences of Bauer，s theorem 132

8.9. The residues of 2p-1 and（p-1）！to modulus p2 135

Ⅸ. THE REPRESENTATION OF NUMBERS BY DECIMALS 138

9.1. The decimal associated with a given number 138

9.2. Terminating and recurring decimals 141

9.3. Representation of numbers in other scales 144

9.4. Irrationals defined by decimals 145

9.5. Tests for divisibility 146

9.6. Decimals with the maximum period 147

9.7. Bachet，s problem of the weights 149

9.8. The game of Nim 151

9.9. Integers with missing digits 154

9.10. Sets of measure zero 155

9.11. Decimals with missing digits 157

9.12. Normal numbers 158

9.13. Proof that almost all numbers are normal 160

Ⅹ. CONTINUED FRACTIONS 165

10.1. Finite continued fractions 165

10.2. Convergents to a continued fraction 166

10.3. Continued fractions with positive quotients 168

10.4. Simple continued fractions 169

10.5. The representation of an irreducible rational fraction by a simple continued fraction 170

10.6. The continued fraction algorithm and Euclid，s algorithm 172

10.7. The difference between the fraction and its convergents 175

10.8. Infinite simple continued fractions 177

10.9. The representation of an irrational number by an infinite continued fraction 178

10.10. A lemma 180

10.11. Equivalent numbers 181

10.12. Periodic continued fractions 184

10.13. Some special quadratic surds 187

10.14. The series of Fibonacci and Lucas 190

10.15. Approximation by convergents 194

Ⅺ. APPROXIMATION OF IRRATIONALS BY RATIONALS 198

11.1. Statement of the problem 198

11.2. Generalities concerning the problem 199

11.3. An argument of Dirichlet 201

11.4. Orders of approximation 202

11.5. Algebraic and transcendental numbers 203

11.6. The existence of transcendental numbers 205

11.7. Lionville，s theorem and the construction of transcendental numbers 206

11.8. The measure of the closest approximations to an arbitrary irrational 208

11.9. Another theorem concerning the convergents to a continued fraction 210

11.10. Continued fractions with bounded quotients 212

11.11. Further theorems concerning approximation 216

11.12. Simultaneous approximation 217

11.13. The transcendence of e 218

11.14. The transcendence of πr 223

Ⅻ. THE FUNDAMENTAL THEOREM OF ARITHMETIC IN k（1），k（i），AND k（ρ） 229

12.1. Algebraic numbers and integers 229

12.2. The rational integers，the Gaussian integers，and the integers of k（ρ） 230

12.3. Euclid，s algorithm 231

12.4. Application of Euclid，s algorithm to the fundamental theorem in k（1） 232

12.5. Historical remarks on Euclid，s algorithm and the fundamental theorem 234

12.6. Properties of the Gaussian integers 235

12.7. Primes in k（i） 236

12.8. The fundamental theorem of arithmetic in k（i） 238

12.9. The integers of k（ρ） 241

ⅩⅢ. SOME DIOPHANTINE EQUATIONS 245

13.1. Fermat，s last theorem 245

13.2. The equation x2+y2=z2 245

13.3. The equation x4+y4=z4 247

13.4. The equation x3+y3=z3 248

13.5. The equation x3+y3=3z3 253

13.6. The expression of a rational as a sum of rational cubes 254

13.7. The equation x3+y3+z3=t3 257

ⅩⅣ. QUADRATIC FIELDS（1） 264

14.1. Algebraic fields 264

14.2. Algebraic numbers and integers； primitive polynomials 265

14.3. The general quadratic field k（√m） 267

14.4. Unities and primes 268

14.5. The unities of k（√2） 270

14.6. Fields in which the fundamental theorem is false 273

14.7. Complex Euclidean fields 274

14.8. Real Euclidean fields 276

14.9. Real Euclidean fields（continued） 279

ⅩⅤ QUADRATIC FIELDS（2） 283

15.1. The primes of k（i） 283

15.2. Fermat，s theorem in k（i） 285

15.3. The primes of k（ρ） 286

15.4. The primes of k（√2） and k（√5） 287

15.5. Lucas，s test for the primality of the Mersenne number M4n+3 290

15.6. General remarks on the arithmetic of quadratic fields 293

15.7. Ideals in a quadratic field 295

15.8. Other fields 299

ⅩⅥ. THE ARITHMETICAL FUNCTIONS φ（n），μ（n），d（n），σ（n），r（n） 302

16.1. The function φ（n） 302

16.2. A further proof of Theorem 63 303

16.3. The Mobius function 304

16.4. The Mobius inversion formula 305

16.5. Further inversion formulae 307

16.6. Evaluation of Ramanujan，s sum 308

16.7. The functions d（n） and σk（n） 310

16.8. Perfect numbers 311

16.9. The function r（n） 313

16.10. Proof of the formula for r（n） 315

ⅩⅦ. GENERATING FUNCTIONS OF ARITHMETICAL FUNCTIONS 318

17.1. The generation of arithmetical functions by means of Dirichlet series 318

17.2. The zeta function 320

17.3. The behaviour of ζ（s） when s→1 321

17.4. Multiplication of Dirichlet series 323

17.5. The generating functions of some special arithmetical functions 326

17.6. The analytical interpretation of the Mobius formula 328

17.7. The function A（n） 331

17.8. Further examples of generating functions 334

17.9. The generating function of r（n） 337

17.10. Generating functions of other types 338

ⅩⅧ. THE ORDER OF MAGNITUDE OF ARITHMETICAL FUNCTIONS 342

18.1. The order of d（n） 342

18.2. The average order of d（n） 347

18.3. The order of σ （n） 350

18.4. The order of φ（n） 352

18.5. The average order of φ（n） 353

18.6. The number of squarefree numbers 355

18.7. The order of r（n） 356

ⅩⅨ. PARTITIONS 361

19.1. The general problem of additive arithmetic 361

19.2. Partitions of numbers 361

19.3. The generating function of p（n） 362

19.4. Other generating functions 365

19.5. Two theorems of Euler 366

19.6. Further algebraical identities 369

19.7. Another formula for F（x） 371

19.8. A theorem of Jacobi 372

19.9. Special cases of Jacobi，s identity 375

19.10. Applications of Theorem 353 378

19.11. Elementary proof of Theorem 358 379

19.12. Congruence properties ofp（n） 380

19.13. The Rogers-Ramanujan identities 383

19.14. Proof of Theorems 362 and 363 386

19.15. Ramanujan，s continued fraction 389

ⅩⅩ. THE REPRESENTATION OF A NUMBER BY TWO OR FOUR SQUARES 393

20.1. Waring，s problem： the numbers g（k） and G（k） 393

20.2. Squares 395

20.3. Second proof of Theorem 366 395

20.4. Third and fourth proofs of Theorem 366 397

20.5. The four-square theorem 399

20.6. Quatemions 401

20.7. Preliminary theorems about integral quaternions 403

20.8. The highest common right-hand divisor of two quaternions 405

20.9. Prime quaternions and the proof of Theorem 370 407

20.10. The values of g（2） and G（2） 409

20.11. Lemmas for the third proof of Theorem 369 410

20.12. Third proof of Theorem 369：the number of representations 411

20.13. Representations by a larger number of squares 415

ⅩⅪ. REPRESENTATION BY CUBES AND HIGHER POWERS 419

21.1. Biquadrates 419

21.2. Cubes：the existence of G（3） and g（3） 420

21.3. A bound for g（3） 422

21.4. Higher powers 424

21.5. A lower bound for g（k） 425

21.6. Lower bounds for G（k） 426

21.7. Sums affected with signs：the number v（k） 431

21.8. Upper bounds for v（k） 433

21.9. The problem of Prouhet and Tarry：the number P（k，j） 435

21.10. Evaluation of P（k，j） for particular k and j 437

21.11. Further problems of Diophantine analysis 440

ⅩⅫ. THE SERIES OF PRIMES （3） 451

22.1. The functions ？（x）andψ（x） 451

22.2. Proof that ？（x） and ψ（x） are of order x 453

22.3. Bertrand，s postulate and a ‘formula，for primes 455

22.4. Proof of Theorems 7 and 9 458

22.5. Two formal transformations 460

22.6. An important sum 461

22.7. The sum ∑p-1 and the product П （1 - p-1） 464

22.8. Mertens，s theorem 466

22.9. Proof of Theorems 323 and 328 469

22.10. The number of prime factors of n 471

22.11. The normal order of ω（n） and Ω （n） 473

22.12. A note on round numbers 476

22.13. The normal order of d （n） 477

22.14. Selberg，s theorem 478

22.15. The functions R（x） and V（ξ） 481

22.16. Completion of the proof of Theorems 434，6，and 8 486

22.17. Proof ofTheorem 335 489

22.18. Products of k prime factors 490

22.19. Primes in an interval 494

22.20. A conjecture about the distribution of prime pairs p，p+2 495

ⅩⅩⅢ. KRONECKER，S THEOREM 501

23.1. Kronecker，s theorem in one dimension 501

23.2. Proofs of the one-dimensional theorem 502

23.3. The problem of the reflected ray 505

23.4. Statement of the general theorem 508

23.5. The two forms of the theorem 510

23.6. An illustration 512

23.7. Lettenmeyer，s proof of the theorem 512

23.8. Estermann，s proof of the theorem 514

23.9. Bohr，s proof of the theorem 517

23.10. Uniform distribution 520

ⅩⅩⅣ. GEOMETRY OF NUMBERS 523

24.1. Introduction and restatement of the fundamental theorem 523

24.2. Simple applications 524

24.3. Arithmetical proof of Theorem 448 527

24.4. Best possible inequalities 529

24.5. The best possible inequality for ξ2+η2 530

24.6. The best possible inequality for |ξη| 532

24.7. A theorem concerning non-homogeneous forms 534

24.8. Arithmetical proof of Theorem 455 536

24.9. Tchebotaref，s theorem 537

24.10. A converse of Minkowski，s Theorem 446 540

ⅩⅩⅤ. ELLIPTIC CURVES 549

25.1. The congruent number problem 549

25.2. The addition law on an elliptic curve 550

25.3. Other equations that define elliptic curves 556

25.4. Points of finite order 559

25.5. The group of rational points 564

25.6. The group of points modulo p. 573

25.7. Integer points on elliptic curves 574

25.8. The L-series of an elliptic curve 578

25.9. Points of finite order and modular curves 582

25.10. Elliptic curves and Fermat，s last theorem 586

APPENDIX 593

1. Another formula for pn 593

2. A generalization of Theorem 22 593

3. Unsolved problems concerning primes 594

A LIST OF BOOKS 597

INDEX OF SPECIAL SYMBOLS AND WORDS 601

INDEX OF NAMES 605

GENERAL INDEX 611