哈代数论(英文版·第6版)

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 peoperties of the fundamental lattice　33
3.7. Third proof of Theorems 28 and 29　35
3.8. The Farev 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　3o
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 orooerties of congruences　60
5.4. Linear congruences　60
5.5. Euler's function φ(m)　63
5.6. Aoolications 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. FFRMAT's THEOREM AND ITS CONSEOUENCES　78
6.1. Fermat's theorem　78
6.2. Some properties of binomial coefficients　79
6.3. A second proof of Theorem 72　81
6.4. Proof of Theorem 22　82
6.5. Quadratic residues　83
6.6. Soecial 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)　IslS
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 orimalitv　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 c~nm'uences to a orime 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 orime-oower modulus　123
8.4. Examoles　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. Convements 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 simplecontinued 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 infinitecontinued fraction　178
10.10. A lemma　180
10.11. Equivalent numbers　181
10.12. Periodic continued fractions　184
10.13. Some soecial 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 oroblem　198
11.2. Generalities concerning the problem　199
11.3. An argument of Dirichlet　201
11.4. Orders of aporoximation　202
11.5. Aloohrnie nncl transcendental numbers　203
11.6. The existence of transcendental numbers　..205
11.7. Liouville'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 π　223
XlI. THE FUNDAMENIAL THEOREM OF ARITHMETIC INk(1), k(i), AND k(O)　229
12.1. Algebraic numbers and integers　229
12.2. The rational integers, the Gaussian integers, and the integers of k(p)　230
12.3. Euclid's algorithm　231
12.4. Aoolication 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. Prooerties of the Gaussian integers　235
12.7. Primes in k(i)　236
12.8. The fundnmental theorem of arithmetic in k(i)　238
12.9. The integers of k(p)　241
XIII. SOME DIOPHANTINE EQUATIONS　245
13.1. Fermat's last theorem　245
13.2. The eauation xz 4- vz = zz　245
13.3. The equation x4 -t- y4 = z4　247
13.4. The equation x3 + y3 = z3　248
13.5. The equation x3 +y3 =3z3　253
13.6. The exoression of a rational as a sum of rational cubes　254
13.7. The equation x3 +y3 +z3 =t3　257
XIV. OUADRATIC FIELDS (1)　264
14.1. Algebraic fields　264
14.2. Algebraic numbers and integers: orimitive polynomials　265
14.3. The general quadratic field k(√m)　267
14.4. Unities and orimes　268
14.5. The unities of k(√2)　270
14.6. Fields in which the fundamental theorem is false　273
14.7. Comnlex Euclidean fields　274
14.8. Real Euclidean fields　276
14.9. Real Euclidean fields (continued)　279
XV. OUADRATIC FIELDS (2)　283
15.1. The orimes of k(i)　283
15.2. Fermat's theorem in k(i)　285
15.3. The primes of k (p)　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 Ф(n),μ(n), d(n), σ(n), r(n) 302
16.1. The function Ф(n)　302
16.2. A further proof of Theorem 63　303
16.3. The Mrbius function　304
16.4. The Mrbius inversion formula　305
16.5. Further inversion formulae　307
16.6. Evaluation of Ramanuian's sum　308
16.7. The functions d(n) and crk (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 ξ(s) when s→1　321
17.4. Multiplication of Dirichlet series　323
17.5. The generating functions of some special arithmetical functions　32~
17.6. The analytical interpretation of the M6bius 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 Ф(n)　352
18.5. The average order of Ф(n)　353
18.6. The number of squarefree numbers　355
18.7. The order of σ(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 ofp(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 of p(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. Quaternions　401
20.7. Preliminary theorems about integral quatemions　403
20.8. The highest common fight-hand divisor of two quatemions　405
20.9. Prime quatemions 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 andj　437
21.11. Further problems of Diophantine analysis　440
XXII. THE SERIES OF PRIMES(3)　451
22.1. The functions 0(x) and $(x)　451
22.2. Proof that 0(x) and ~ (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 12p~ 1 and the product FI (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 to (n) and f2 (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 of Theorem 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 + n2　530
24.6. The best possible inequality for

.　　G. H. H.
　　E. M. W.
　　OXFORD ...
　　August 1938