哈代數(shù)論（英文版）

Ⅰ. THE SERIES OF PRIMES (1)
1.1. Divisibility of integers
1.2. Prime numbers
1.3. Statement of the fundamental theorem of arithmetic
1.4. The sequence of primes
1.5. Some questions concerning primes
1.6. Some notations
1.7. The logarithmic function
1.8. Statement of the prime number theorem
Ⅱ. THE SERIES OF PRIMES (2)
2.1. First proof of Euclids second theorem
2.2. Further deductions from Euclids argument
2.3. Primes in certain arithmetical progressions
2.4. Second proof of Euclids theorem
2.5. Fermats and Mersennes numbers
2.6. Third proof of Euclids theorem
2.7. Further results on formulae for primes
2.8. Unsolved problems concerning primes
2.9. Moduli of integers
2.10. Proof of the fundamental theorem or arithmetic
2.11. Another proof of the fundamental theorem
Ⅲ. FAREY SERIES AND A THEOREM OF MINKOWSKI
3.1. The definition and simplest properties of a Farey series
3.2. The equivalence of the two characteristic properties
3.3. First proof of Theorems 28 and
3.4. Second proof of the theorems
3.5. The integral lattice
3.6. Some simple peoperties of the fundamental lattice
3.7. Third proof of Theorems 28 and
3.8. The Farev dissection of the continuum
3.9. A theorem of Minkowski
3.10 Proof of Minkowskis theorem
3.11. Developments of Theorem
Ⅳ. IRRATIONAL NUMBERS
4.1. Some generalities
4.2. Numbers known to be irrational
4.3. The theorem of Pythagoras and its generalizations
4.4. The use of the fundamental theorem in the proofs of Theorems 43-
4.5. A historical digression 3o
4.6. Geometrical proof of the irrationality of √
4.7. Some more irrational numbers
Ⅴ. CONGRUENCES AND RESIDUES
5.1. Highest common divisor and least common multiple
5.2. Congruences and classes of residues
5.3. Elementary orooerties of congruences
5.4. Linear congruences
5.5. Eulers function φ(m)
5.6. Aoolications of Theorems 59 and 61 to trigonometrical sums
5.7. A general principle
5.8. Construction of the regular polygon of 17 sides
Ⅵ. FFRMATs THEOREM AND ITS CONSEOUENCES
6.1. Fermats theorem
6.2. Some properties of binomial coefficients
6.3. A second proof of Theorem
6.4. Proof of Theorem
6.5. Quadratic residues
6.6. Soecial cases of Theorem 79： Wilsons theorem
6.7. Elementary properties of quadratic residues and non-residues
6.8. The order of a (mod m) IslS
6.9. The converse of Fermats theorem
6.10. Divisibility of 2P-1 _ 1 by p
6.11. Gausss lemma and the quadratic character of
6.12. The law of reciprocity
6.13. Proof of the law of reciprocity
6.14. Tests for orimalitv
6.15. Factors of Mersenne numbers; a theorem of Euler
Ⅶ. GENERAL PROPERTIES OF CONGRUENCES
7.1. Roots of congruences
7.2. Integral polynomials and identical congruences
7.3. Divisibility of polynomials (mod m)
7.4. Roots of c~nmuences to a orime modulus
7.5. Some applications of the general theorems
7.6. Lagranges proof of Fermats and Wilsons theorems
7.7. The residue of { 1/2 (p - 1 ) }!
7.8. A theorem of Wolstenholme
7.9. The theorem of von Staudt
7.10. Proof of von Staudts theorem
Ⅷ. CONGRUENCES TO COMPOSITE MODULI
8.1. Linear congruences
8.2. Congruences of higher degree
8.3. Congruences to a orime-oower modulus
8.4. Examoles
8.5. Bauers identical congruence
8.6. Bauers congruence： the case p=
8.7. A theorem of Leudesdorf
8.8. Further consequences of Bauers theorem
8.9. The residues of 2P-1 and (p - 1)! to modulus p
Ⅸ. THE REPRESENTATION OF NUMBERS BY DECIMALS
9.1. The decimal associated with a given number
9.2. Terminating and recurring decimals
9.3. Representation of numbers in other scales
9.4. Irrationals defined by decimals
9.5. Tests for divisibility
9.6. Decimals with the maximum period
9.7. Bachets problem of the weights
9.8. The game of Nim
9.9. Integers with missing digits
9.10. Sets of measure zero
9.11. Decimals with missing digits
9.12. Normal numbers
9.13. Proof that almost all numbers are normal
Ⅹ. CONTINUED FRACTIONS
10.1. Finite continued fractions
10.2. Convements to a continued fraction
10.3. Continued fractions with positive quotients
10.4. Simple continued fractions
10.5. The representation of an irreducible rational fraction by a simplecontinued fraction
10.6. The continued fraction algorithm and Euclids algorithm
10.7. The difference between the fraction and its convergents
10.8. Infinite simple continued fractions
10.9. The representation of an irrational number by an infinitecontinued fraction
10.10. A lemma
10.11. Equivalent numbers
10.12. Periodic continued fractions
10.13. Some soecial Quadratic surds
10.14. The series of Fibonacci and Lucas
10.15. Approximation by convergents
Ⅺ. APPROXIMATION OF IRRATIONALS BY RATIONALS
11.1. Statement of the oroblem
11.2. Generalities concerning the problem
11.3. An argument of Dirichlet
11.4. Orders of aporoximation
11.5. Aloohrnie nncl transcendental numbers
11.6. The existence of transcendental numbers ..
11.7. Liouvilles theorem and the construction of transcendental numbers
11.8. The measure of the closest approximations to an arbitrary irrational
11.9. Another theorem concerning the convergents to a continued fraction
11.10. Continued fractions with bounded quotients
11.11. Further theorems concerning approximation
11.12. Simultaneous approximation
11.13. The transcendence of e
11.14. The transcendence of π
Ⅻ. THE FUNDAMENIAL THEOREM OF ARITHMETIC INk(1)， k(i)， AND k(O)
12.1. Algebraic numbers and integers
12.2. The rational integers， the Gaussian integers， and the integers of k(p)
12.3. Euclids algorithm
12.4. Aoolication of Euclids algorithm to the fundamental theorem in k(1)
12.5. Historical remarks on Euclids algorithm and the fundamental theorem
12.6. Prooerties of the Gaussian integers
12.7. Primes in k(i)
12.8. The fundnmental theorem of arithmetic in k(i)
12.9. The integers of k(p)
ⅩⅢ. SOME DIOPHANTINE EQUATIONS
13.1. Fermats last theorem
13.2. The eauation xz 4- vz = zz
13.3. The equation x4 -t- y4 = z
13.4. The equation x3 + y3 = z
13.5. The equation x3 +y3 =3z
13.6. The exoression of a rational as a sum of rational cubes
13.7. The equation x3 +y3 +z3 =t
ⅩⅣ. OUADRATIC FIELDS (1)
14.1. Algebraic fields
14.2. Algebraic numbers and integers： orimitive polynomials
14.3. The general quadratic field k(√m)
14.4. Unities and orimes
14.5. The unities of k(√2)
14.6. Fields in which the fundamental theorem is false
14.7. Comnlex Euclidean fields
14.8. Real Euclidean fields
14.9. Real Euclidean fields (continued)
ⅩⅤ. OUADRATIC FIELDS (2)
15.1. The orimes of k(i)
15.2. Fermats theorem in k(i)
15.3. The primes of k (p)
15.4. The primes of k(√2) and k(√5)
15.5. Lucass test for the primality of the Mersenne number M4n+
15.6. General remarks on the arithmetic of quadratic fields
15.7. Ideals in a quadratic field
15.8. Other fields
ⅩⅥ. THE ARITHMETICAL FUNCTIONS Ф(n)，μ(n)， d(n)， σ(n)， r(n)
16.1. The function Ф(n)
16.2. A further proof of Theorem
16.3. The Mrbius function
16.4. The Mrbius inversion formula
16.5. Further inversion formulae
16.6. Evaluation of Ramanuians sum
16.7. The functions d(n) and crk (n)
16.8. Perfect numbers
16.9. The function r(n)
16.10. Proof of the formula for r(n)
ⅩⅦ. GENERATING FUNCTIONS OF ARITHMETICAL FUNCTIONS
17.1. The generation of arithmetical functions by means of Dirichlet series
17.2. The zeta function
17.3. The behaviour of ξ(s) when s→
17.4. Multiplication of Dirichlet series
17.5. The generating functions of some special arithmetical functions 32~
17.6. The analytical interpretation of the M6bius formula
17.7. The function A(n)
17.8. Further examples of generating functions
17.9. The generating function of r(n)
17.10. Generating functions of other types
ⅩⅧ. THE ORDER OF MAGNITUDE OF ARITHMETICAL FUNCTIONS
18.1. The order of d(n)
18.2. The average order of d(n)
18.3. The order of σ(n)
18.4. The order of Ф(n)
18.5. The average order of Ф(n)
18.6. The number of squarefree numbers
18.7. The order of σ(n)
ⅩⅨ. PARTITIONS
19.1. The general problem of additive arithmetic
19.2. Partitions of numbers
19.3. The generating function ofp(n)
19.4. Other generating functions
19.5. Two theorems of Euler
19.6. Further algebraical identities
19.7. Another formula for F(x)
19.8. A theorem of Jacobi
19.9. Special cases of Jacobis identity
19.10. Applications of Theorem
19.11. Elementary proof of Theorem
19.12. Congruence properties of p(n)
19.13. The Rogers-Ramanujan identities
19.14. Proof of Theorems 362 and
19.15. Ramanujans continued fraction
ⅩⅩ. THE REPRESENTATION OF A NUMBER BY TWO OR FOUR SQUARES
20.1. Warings problem： the numbers g(k) and G(k)
20.2. Squares
20.3. Second proof of Theorem
20，4. Third and fourth proofs of Theorem
20.5. The four-square theorem
20.6. Quaternions
20.7. Preliminary theorems about integral quatemions
20.8. The highest common fight-hand divisor of two quatemions
20.9. Prime quatemions and the proof of Theorem
20.10. The values of g(2) and G(2)
20.11. Lemmas for the third proof of Theorem
20.12. Third proof of Theorem 369： the number of representations
20.13. Representations by a larger number of squares
ⅩⅩⅠ. REPRESENTATION BY CUBES AND HIGHER POWERS
21.1. Biquadrates
21.2. Cubes： the existence of G(3) and g(3)
21.3. A bound for g(3)
21.4. Higher powers
21.5. A lower bound for g(k)
21.6. Lower bounds for G(k)
21.7. Sums affected with signs： the number v(k)
21.8. Upper bounds for v(k)
21.9. The problem of Prouhet and Tarry： the number P(k，j)
21.10. Evaluation of P(k，j) for particular k andj
21.11. Further problems of Diophantine analysis
ⅩⅩⅡ. THE SERIES OF PRIMES(3)
22.1. The functions 0(x) and $(x)
22.2. Proof that 0(x) and ~ (x) are of order x
22.3. Bertrands postulate and a formula for primes
22.4. Proof of Theorems 7 and
22.5. Two formal transformations
22.6. An important sum
22.7. The sum 12p~ 1 and the product FI (1 - p- 1 )
22.8. Mertenss theorem
22.9. Proof of Theorems 323 and
22.10. The number of prime factors of n
22.11. The normal order of to (n) and f2 (n)
22.12. A note on round numbers
22.13. The normal order of d(n)
22.14. Selbergs theorem
22.15. The functions R (x) and V (ξ)
22.16. Completion of the proof of Theorems 434， 6， and
22.17. Proof of Theorem
22.18. Products of k prime factors
22.19. Primes in an interval
22.20. A conjecture about the distribution of prime pairs p， p +
ⅩⅩⅢ. KRONECKERS THEOREM
23.1. Kroneckers theorem in one dimension
23.2. Proofs of the one-dimensional theorem
23.3. The problem of the reflected ray
23.4. Statement of the general theorem
23.5. The two forms of the theorem
23.6. An illustration
23.7. Lettenmeyers proof of the theorem
23.8. Estermanns proof of the theorem
23.9. Bohrs proof of the theorem
23.10. Uniform distribution
ⅩⅩⅣ. GEOMETRY OF NUMBERS
24.1. Introduction and restatement of the fundamental theorem
24.2. Simple applications
24.3. Arithmetical proof of Theorem
24.4. Best possible inequalities
24.5. The best possible inequality for ξ2 + n
24.6. The best possible inequality for |ξn|
24.7. A theorem concerning non-homogeneous forms
24.8. Arithmetical proof of Theorem
24.9. Tchebotarefs theorem
24.10. A converse of Minkowskis Theorem
ⅩⅩⅤ. ELLIPTIC CURVES
25.1. The congruent number problem
25.2. The addition law on an elliptic curve
25.3. Other equations that define elliptic curves
25.4. Points of finite order
25.5. The group of rational points
25.6. The group of points modulo p.
25.7. Integer points on elliptic curves
25.8. The L-series of an elliptic curve
25.9. Points of finite order and modular curves
25.10. Elliptic curves and Fermats last theorem
APPENDIX
1. Another formula forpn
2. A generalization of Theorem
3. Unsolved problems concerning primes
A LIST OF BOOKS
INDEX OF SPECIAL SYMBOLS AND WORDS
INDEX OF NAMES
GENERAL INDEX