哈代数论 英文版2025|PDF|Epub|mobi|kindle电子书版本百度云盘下载

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Ⅰ. THE SERIES OF PRIMES（1）1

1.1. Divisibility of integers1

1.2. Prime numbers2

1.3. Statement of the fundamental theorem of arithmetic3

1.4. The sequence of primes4

1.5. Some questions concerning primes6

1.6. Some notations7

1.7. The logarithmic function9

1.8. Statement of the prime number theorem10

Ⅱ. THE SERIES OF PRIMES（2）14

2.1. First proof of Euclid，s second theorem14

2.2. Further deductions from Euclid，s argument14

2.3. Primes in certain arithmetical progressions15

2.4. Second proof of Euclid，s theorem17

2.5. Fermat，s and Mersenne，s numbers18

2.6. Third proof of Euclid，s theorem20

2.7. Further results on formulae for primes21

2.8. Unsolved problems concerning primes23

2.9. Moduli of integers23

2.10. Proof of the fundamental theorem of arithmetic25

2.11. Another proof of the fundamental theorem26

Ⅲ. FAREY SERIES AND A THEOREM OF MINKOWSKI28

3.1. The definition and simplest properties of a Farey series28

3.2. The equivalence of the two characteristic properties29

3.3. First proof of Theorems 28 and 2930

3.4. Second proof of the theorems31

3.5. The integral lattice32

3.6. Some simple properties of the fundamental lattice33

3.7. Third proof of Theorems 28 and 2935

3.8. The Farey dissection of the continuum36

3.9. A theorem of Minkowski37

3.10. Proof of Minkowski，s theorem39

3.11. Developments of Theorem 3740

Ⅳ. IRRATIONAL NUMBERS45

4.1. Some generalities45

4.2. Numbers known to be irrational46

4.3. The theorem of Pythagoras and its generalizations47

4.4. The use of the fundamental theorem in the proofs of Theorems 43-4549

4.5. A historical digression50

4.6. Geometrical proof of the irrationality of √552

4.7. Some more irrational numbers53

Ⅴ. CONGRUENCES AND RESIDUES57

5.1. Highest common divisor and least common multiple57

5.2. Congruences and classes of residues58

5.3. Elementary properties of congruences60

5.4. Linear congruences60

5.5. Euler，s function φ（m）63

5.6. Applications of Theorems 59 and 61 to trigonometrical sums65

5.7. A general principle70

5.8. Construction of the regular polygon of 17 sides71

Ⅵ. FERMAT，S THEOREM AND ITS CONSEQUENCES78

6.1. Fermat，s theorem78

6.2. Some properties of binomial coeffcients79

6.3. A second proof of Theorem 7281

6.4. Proof of Theorem 2282

6.5. Quadratic residues83

6.6. Special cases of Theorem 79： Wilson，s theorem85

6.7. Elementary properties of quadratic residues and non-residues87

6.8. The order of a （mod m）88

6.9. The converse of Fermat，s theorem89

6.10. Divisibility of 2p-1 -1 by p291

6.11. Gauss，s lemma and the quadratic character of 292

6.12. The law of reciprocity95

6.13. Proof of the law of reciprocity97

6.14. Tests for primality98

6.15. Factors of Mersenne numbers； a theorem of Euler100

Ⅶ. GENERAL PROPERTIES OF CONGRUENCES103

7.1. Roots of congruences103

7.2. Integral polynomials and identical congruences103

7.3. Divisibility of polynomials （mod m）105

7.4. Roots of congruences to a prime modulus106

7.5. Some applications of the general theorems108

7.6. Lagrange，s proof of Fermat，s and Wilson，s theorems110

7.7. The residue of ｛1/2（p-1）｝！111

7.8. A theorem of Wolstenholme112

7.9. The theorem of von Staudt115

7.10. Proof of von Staudt，s theorem116

Ⅷ. CONGRUENCES TO COMPOSITE MODULI120

8.1. Linear congruences120

8.2. Congruences of higher degree122

8.3. Congruences to a prime-power modulus123

8.4. Examples125

8.5. Bauer，s identical congruence126

8.6. Bauer，s congruence： the case p=2129

8.7. A theorem of Leudesdorf130

8.8. Further consequences of Bauer，s theorem132

8.9. The residues of 2p-1 and（p-1）！to modulus p2135

Ⅸ. THE REPRESENTATION OF NUMBERS BY DECIMALS138

9.1. The decimal associated with a given number138

9.2. Terminating and recurring decimals141

9.3. Representation of numbers in other scales144

9.4. Irrationals defined by decimals145

9.5. Tests for divisibility146

9.6. Decimals with the maximum period147

9.7. Bachet，s problem of the weights149

9.8. The game of Nim151

9.9. Integers with missing digits154

9.10. Sets of measure zero155

9.11. Decimals with missing digits157

9.12. Normal numbers158

9.13. Proof that almost all numbers are normal160

Ⅹ. CONTINUED FRACTIONS165

10.1. Finite continued fractions165

10.2. Convergents to a continued fraction166

10.3. Continued fractions with positive quotients168

10.4. Simple continued fractions169

10.5. The representation of an irreducible rational fraction by a simple continued fraction170

10.6. The continued fraction algorithm and Euclid，s algorithm172

10.7. The difference between the fraction and its convergents175

10.8. Infinite simple continued fractions177

10.9. The representation of an irrational number by an infinite continued fraction178

10.10. A lemma180

10.11. Equivalent numbers181

10.12. Periodic continued fractions184

10.13. Some special quadratic surds187

10.14. The series of Fibonacci and Lucas190

10.15. Approximation by convergents194

Ⅺ. APPROXIMATION OF IRRATIONALS BY RATIONALS198

11.1. Statement of the problem198

11.2. Generalities concerning the problem199

11.3. An argument of Dirichlet201

11.4. Orders of approximation202

11.5. Algebraic and transcendental numbers203

11.6. The existence of transcendental numbers205

11.7. Lionville，s theorem and the construction of transcendental numbers206

11.8. The measure of the closest approximations to an arbitrary irrational208

11.9. Another theorem concerning the convergents to a continued fraction210

11.10. Continued fractions with bounded quotients212

11.11. Further theorems concerning approximation216

11.12. Simultaneous approximation217

11.13. The transcendence of e218

11.14. The transcendence of πr223

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

12.1. Algebraic numbers and integers229

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

12.3. Euclid，s algorithm231

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 theorem234

12.6. Properties of the Gaussian integers235

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 EQUATIONS245

13.1. Fermat，s last theorem245

13.2. The equation x2+y2=z2245

13.3. The equation x4+y4=z4247

13.4. The equation x3+y3=z3248

13.5. The equation x3+y3=3z3253

13.6. The expression of a rational as a sum of rational cubes254

13.7. The equation x3+y3+z3=t3257

ⅩⅣ. QUADRATIC FIELDS（1）264

14.1. Algebraic fields264

14.2. Algebraic numbers and integers； primitive polynomials265

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

14.4. Unities and primes268

14.5. The unities of k（√2）270

14.6. Fields in which the fundamental theorem is false273

14.7. Complex Euclidean fields274

14.8. Real Euclidean fields276

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+3290

15.6. General remarks on the arithmetic of quadratic fields293

15.7. Ideals in a quadratic field295

15.8. Other fields299

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 63303

16.3. The Mobius function304

16.4. The Mobius inversion formula305

16.5. Further inversion formulae307

16.6. Evaluation of Ramanujan，s sum308

16.7. The functions d（n） and σk（n）310

16.8. Perfect numbers311

16.9. The function r（n）313

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

ⅩⅦ. GENERATING FUNCTIONS OF ARITHMETICAL FUNCTIONS318

17.1. The generation of arithmetical functions by means of Dirichlet series318

17.2. The zeta function320

17.3. The behaviour of ζ（s） when s→1321

17.4. Multiplication of Dirichlet series323

17.5. The generating functions of some special arithmetical functions326

17.6. The analytical interpretation of the Mobius formula328

17.7. The function A（n）331

17.8. Further examples of generating functions334

17.9. The generating function of r（n）337

17.10. Generating functions of other types338

ⅩⅧ. THE ORDER OF MAGNITUDE OF ARITHMETICAL FUNCTIONS342

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 numbers355

18.7. The order of r（n）356

ⅩⅨ. PARTITIONS361

19.1. The general problem of additive arithmetic361

19.2. Partitions of numbers361

19.3. The generating function of p（n）362

19.4. Other generating functions365

19.5. Two theorems of Euler366

19.6. Further algebraical identities369

19.7. Another formula for F（x）371

19.8. A theorem of Jacobi372

19.9. Special cases of Jacobi，s identity375

19.10. Applications of Theorem 353378

19.11. Elementary proof of Theorem 358379

19.12. Congruence properties ofp（n）380

19.13. The Rogers-Ramanujan identities383

19.14. Proof of Theorems 362 and 363386

19.15. Ramanujan，s continued fraction389

ⅩⅩ. THE REPRESENTATION OF A NUMBER BY TWO OR FOUR SQUARES393

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

20.2. Squares395

20.3. Second proof of Theorem 366395

20.4. Third and fourth proofs of Theorem 366397

20.5. The four-square theorem399

20.6. Quatemions401

20.7. Preliminary theorems about integral quaternions403

20.8. The highest common right-hand divisor of two quaternions405

20.9. Prime quaternions and the proof of Theorem 370407

20.10. The values of g（2） and G（2）409

20.11. Lemmas for the third proof of Theorem 369410

20.12. Third proof of Theorem 369：the number of representations411

20.13. Representations by a larger number of squares415

ⅩⅪ. REPRESENTATION BY CUBES AND HIGHER POWERS419

21.1. Biquadrates419

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

21.3. A bound for g（3）422

21.4. Higher powers424

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 j437

21.11. Further problems of Diophantine analysis440

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 x453

22.3. Bertrand，s postulate and a ‘formula，for primes455

22.4. Proof of Theorems 7 and 9458

22.5. Two formal transformations460

22.6. An important sum461

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

22.8. Mertens，s theorem466

22.9. Proof of Theorems 323 and 328469

22.10. The number of prime factors of n471

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

22.12. A note on round numbers476

22.13. The normal order of d （n）477

22.14. Selberg，s theorem478

22.15. The functions R（x） and V（ξ）481

22.16. Completion of the proof of Theorems 434，6，and 8486

22.17. Proof ofTheorem 335489

22.18. Products of k prime factors490

22.19. Primes in an interval494

22.20. A conjecture about the distribution of prime pairs p，p+2495

ⅩⅩⅢ. KRONECKER，S THEOREM501

23.1. Kronecker，s theorem in one dimension501

23.2. Proofs of the one-dimensional theorem502

23.3. The problem of the reflected ray505

23.4. Statement of the general theorem508

23.5. The two forms of the theorem510

23.6. An illustration512

23.7. Lettenmeyer，s proof of the theorem512

23.8. Estermann，s proof of the theorem514

23.9. Bohr，s proof of the theorem517

23.10. Uniform distribution520

ⅩⅩⅣ. GEOMETRY OF NUMBERS523

24.1. Introduction and restatement of the fundamental theorem523

24.2. Simple applications524

24.3. Arithmetical proof of Theorem 448527

24.4. Best possible inequalities529

24.5. The best possible inequality for ξ2+η2530

24.6. The best possible inequality for |ξη|532

24.7. A theorem concerning non-homogeneous forms534

24.8. Arithmetical proof of Theorem 455536

24.9. Tchebotaref，s theorem537

24.10. A converse of Minkowski，s Theorem 446540

ⅩⅩⅤ. ELLIPTIC CURVES549

25.1. The congruent number problem549

25.2. The addition law on an elliptic curve550

25.3. Other equations that define elliptic curves556

25.4. Points of finite order559

25.5. The group of rational points564

25.6. The group of points modulo p.573

25.7. Integer points on elliptic curves574

25.8. The L-series of an elliptic curve578

25.9. Points of finite order and modular curves582

25.10. Elliptic curves and Fermat，s last theorem586

APPENDIX593

1. Another formula for pn593

2. A generalization of Theorem 22593

3. Unsolved problems concerning primes594

A LIST OF BOOKS597

INDEX OF SPECIAL SYMBOLS AND WORDS601

INDEX OF NAMES605

GENERAL INDEX611