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El. knyga: Introduction to Analytic Number Theory

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"This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory. For this reason, the book starts with the most elementary properties of the natural integers. Nevertheless, the text succeeds in presenting an enormous amount of material in little more than 300 pages."--MATHEMATICAL REVIEWS

Recenzijos

From the reviews:

T.M. Apostol

Introduction to Analytic Number Theory

"This book is the first volume of a two-volume textbook for undergraduates and is indeed the crystallization of a course offered by the author at the California Institute of Technology to undergraduates without any previous knowledge of number theory. For this reason, the book starts with the most elementary properties of the natural integers. Nevertheless, the text succeeds in presenting an enormous amount of material in little more than 300 pages. The presentation is invariably lucid and the book is a real pleasure to read."

MATHEMATICAL REVIEWS

After reading Introduction to Analytic Number Theory one is left with the impression that the author, Tom M. Apostal, has pulled off some magic trick. I must admit that I love this book. The selection of topics is excellent, the exposition is fluid, the proofs are clear and nicely structured, and every chapter contains itsown set of exercises. this book is very readable and approachable, and it would work very nicely as a text for a second course in number theory. (Įlvaro Lozano-Robledo, The Mathematical Association of America, December, 2011)

Daugiau informacijos

Springer Book Archives
Historical Introduction
Chapter 1 The Fundamental Theorem of Arithmetic
1.1 Introduction
13(1)
1.2 Divisibility
14(1)
1.3 Greatest common divisor
14(2)
1.4 Prime numbers
16(1)
1.5 The fundamental theorem of arithmetic
17(1)
1.6 The series of reciprocals of the primes
18(1)
1.7 The Euclidean algorithm
19(1)
1.8 The greatest common divisor of more than two numbers
20(4)
Exercises for
Chapter 1
21(3)
Chapter 2 Arithmetical Functions and Dirichlet Multiplication
2.1 Introduction
24(1)
2.2 The Mobius function μ(n)
24(1)
2.3 The Euler totient function φ(n)
25(1)
2.4 A relation connecting φ and μ
26(1)
2.5 A product formula for φ(n)
27(2)
2.6 The Dirichlet product of arithmetical functions
29(1)
2.7 Dirichlet inverses and the Mobius inversion formula
30(2)
2.8 The Mangoldt function Λ(n)
32(1)
2.9 Multiplicative functions
33(2)
2.10 Multiplicative functions and Dirichlet multiplication
35(1)
2.11 The inverse of a completely multiplicative function
36(21)
2.12 Liouville's function λ(n)
57
2.13 The divisor functions σ,(n)
38(1)
2.14 Generalized convolutions
39(2)
2.15 Formal power series
41(1)
2.16 The Bell series of an arithmetical function
42(2)
2.17 Bell series and Dirichlet multiplication
44(1)
2.18 Derivatives of arithmetical functions
45(1)
2.19 The Selberg identity
46(6)
Exercises for
Chapter 2
46(6)
Chapter 3 Averages of Arithmetical Functions
3.1 Introduction
52(1)
3.2 The big oh notation. Asymptotic equality of functions
53(1)
3.3 Euler's summation formula
54(1)
3.4 Some elementary asymptotic formulas
55
3.5 The average order of d(n)
51(9)
3.6 The average order of the divisor functions σx(n)
60(1)
3.7 The average order of φ(n)
61(1)
3.8 An application to the distribution of lattice points visible from the origin
62(2)
3.9 The average order of μ(n) and of Λ(n)
64(1)
3.10 The partial sums of a Dirichlet product
65(1)
3.11 Applications to p(n) and λ(n)
66(3)
3.12 Another identity for the partial sums of a Dirichlet product
69
Exercises for
Chapter 3
70
Chapter 4 Some Elementary Theorems on the Distribution of Prime Numbers
4.1 Introduction
14(61)
4.2 Chebyshev's functions ψ(x) and π(x)
75
4.3 Relations connecting (x) and π(x)
16(3)
4.4 Some equivalent forms of the prime number theorem
19(63)
4.5 Inequalities for n(n) and ρn
82(3)
4.6 Shapiro's Tauberian theorem
85(3)
4.7 Applications of Shapiro's theorem
88(1)
4.8 An asymptotic formula for the partial sums Σp≤x (1/p)
89(2)
4.9 The partial sums of the Mobius function
91(7)
4.10 Brief sketch of an elementary proof of the prime number theorem
98(1)
4.11 Selberg's asymptotic formula
99(7)
Exercises for
Chapter 4
101(5)
Chapter 5 Congruences
5.1 Definition and basic properties of congruences
106(3)
5.2 Residue classes and complete residue systems
109(1)
5.3 Linear congruences
110(3)
5.4 Reduced residue systems and the Euler-Fermat theorem
113(1)
5.5 Polynomial congruences modulo ρ. Lagrange's theorem
114(1)
5.6 Applications of Lagrange's theorem
115(2)
5.7 Simultaneous linear congruences. The Chinese remainder theorem
117(1)
5.8 Applications of the Chinese remainder theorem
118(2)
5.9 Polynomial congruences with prime power moduli
120(3)
5.10 The principle of cross-classification
123(2)
5.11 A decomposition property of reduced residue systems
125(4)
Exercises for
Chapter 5
126(3)
Chapter 6 Finite Abelian Groups and Their Characters
6.1 Definitions
129(1)
6.2 Examples of groups and subgroups
130(1)
6.3 Elementary properties of groups
130(1)
6.4 Construction of subgroups
131(2)
6.5 Characters of finite abelian groups
133(2)
6.6 The character group
135(1)
6.7 The orthogonality relations for characters
136(1)
6.8 Dirichlet characters
137(3)
6.9 Sums involving Dirichlet characters
140(1)
6.10 The non vanishing of L(L χ) for real nonprincipal χ
141(5)
Exercises for
Chapter 6
143(3)
Chapter 7 Dirichlet's Theorem on Primes in Arithmetic Progressions
7.1 Introduction
146(1)
7.2 Dirichlet's theorem for primes of the form 4n -- 1 and 4n + 1
147(1)
7.3 The plan of the proof of Dirichlet's theorem
148(2)
7.4 Proof of Lemma 7.4
150(1)
7.5 Proof of Lemma 7.5
151(1)
7.6 Proof of Lemma 7.6
152(1)
1.1 Proof of Lemma 7.8
153(1)
7.8 Proof of Lemma 7.7
153(1)
1.9 Distribution of primes in arithmetic progressions
154(3)
Exercises for
Chapter 7
155(2)
Chapter 8 Periodic Arithmetical Functions and Gauss Sums
8.1 Functions periodic modulo k
157(1)
8.2 Existence of finite Fourier series for periodic arithmetical functions
158(2)
8.3 Ramanujan's sum and generalizations
160(2)
8.4 Multiplicative properties of the sums sk(n)
162(3)
8.5 Gauss sums associated with Dirichlet characters
165(1)
8.6 Dirichlet characters with non vanishing Gauss sums
166(1)
8.7 Induced moduli and primitive characters
167(1)
8.8 Further properties of induced moduli
168(3)
8.9 The conductor of a character
171(1)
8.10 Primitive characters and separable Gauss sums
171(1)
8.11 The finite Fourier series of the Dirichlet characters
172(1)
8.12 Polya's inequality for the partial sums of primitive characters
173(5)
Exercises for
Chapter 8
175(3)
Chapter 9 Quadratic Residues and the Quadratic Reciprocity Law
9.1 Quadratic residues
178(1)
9.2 Legendre's symbol and its properties
179(2)
9.3 Evaluation of (--1|ρ and (2|ρ)
181(1)
9.4 Gauss' lemma
182(3)
9.5 The quadratic reciprocity law
185(1)
9.6 Applications of the reciprocity law
186(1)
9.7 The Jacobi symbol
187(3)
9.8 Applications to Diophantine equations
190(2)
9.9 Gauss sums and the quadratic reciprocity law
192(3)
9.10 The reciprocity law for quadratic Gauss sums
195(5)
9.11 Another proof of the quadratic reciprocity law
200(4)
Exercises for
Chapter 9
201(3)
Chapter 10 Primitive Roots
10.1 The exponent of a number mod m. Primitive roots
204(1)
10.2 Primitive roots and reduced residue systems
205(1)
10.3 The nonexistence of primitive roots mod 2x for α ≥ 3
206(1)
10.4 The existence of primitive roots mod ρ for odd primes ρ
206(2)
10.5 Primitive roots and quadratic residues
208(1)
10.6 The existence of primitive roots mod ρx
208(2)
10.7 The existence of primitive roots mod 2ρx
210(1)
10.8 The nonexistence of primitive roots in the remaining cases
211(1)
10.9 The number of primitive roots mod m
212(1)
10.10 The index calculus
213(5)
10.11 Primitive roots and Dirichlet characters
218(2)
10.12 Real-valued Dirichlet characters mod ρx
220(1)
10.13 Primitive Dirichlet characters mod ρx
221(3)
Exercises for
Chapter 10
222(2)
Chapter 11 Dirichlet Series and Euler Products
11.1 Introduction
224(1)
11.2 The half-plane of absolute convergence of a Dirichlet series
225(1)
11.3 The function defined by a Dirichlet series
226(2)
11.4 Multiplication of Dirichlet series
228(2)
11.5 Euler products
230(2)
11.6 The half-plane of convergence of a Dirichlet series
232(2)
11.7 Analytic properties of Dirichlet series
234(2)
11.8 Dirichlet series with nonnegative coefficients
236(2)
11.9 Dirichlet series expressed as exponentials of Dirichlet series
238(2)
11.10 Mean value formulas for Dirichlet series
240(2)
11.11 An integral formula for the coefficients of a Dirichlet series
242(1)
11.12 An integral formula for the partial sums of a Dirichlet series
243(6)
Exercises for
Chapter 11
246(3)
Chapter 12 The Functions ζ(s) and L(s, χ)
12.1 Introduction
249(1)
12.2 Properties of the gamma function
250(1)
12.3 Integral representation for the Hurwitz zeta function
251(2)
12.4 A contour integral representation for the Hurwitz zeta function
253(1)
12.5 The analytic continuation of the Hurwitz zeta function
254(1)
12.6 Analytic continuation of ζ(s) and L(s, χ)
255(1)
12.7 Hurwitz's formula for ζ(s, a)
256(3)
12.8 The functional equation for the Riemann zeta function
259(2)
12.9 A functional equation for the Hurwitz zeta function
261(1)
12.10 The functional equation for L-functions
261(3)
12.11 Evaluation of ζ(n, a)
264(1)
12.12 Properties of Bernoulli numbers and Bernoulli polynomials
265(3)
12.13 Formulas for L(O, χ)
268(1)
12.14 Approximation of ζs, a) by finite sums
268(2)
12.15 Inequalities for |ζ(s, a)
270(2)
12.16 Inequalities for |ζ(s)| and |L(s, χ)|
272(6)
Exercises for
Chapter 12
273(5)
Chapter 13 Analytic Proof of the Prime Number Theorem
13.1 The plan of the proof
278(1)
13.2 Lemmas
279(4)
13.3 A contour integral representation for ψ1(x)/x2
283(1)
13.4 Upper bounds for |ζ(s)| and |ζ(s)| near the line σ = 1
284(2)
13.5 The nonvanishing of ζ(s) on the line σ = 1
286(1)
13.6 Inequalities for |1/ζ(s)| and |ζ(s)/ζ(s)|
287(2)
13.7 Completion of the proof of the prime number theorem
289(2)
13.8 Zero-free regions for ζ(s)
291(2)
13.9 The Riemann hypothesis
293(1)
13.10 Application to the divisor function
294(3)
13.11 Application to Euler's totient
297(2)
13.12 Extension of Polya's inequality for character sums
299(5)
Exercises for
Chapter 13
300(4)
Chapter 14 Partitions
14.1 Introduction
304(3)
14.2 Geometric representation of partitions
307(1)
14.3 Generating functions for partitions
308(3)
14.4 Euler's pentagonal-number theorem
311(2)
14.5 Combinatorial proof of Euler's pentagonal-number theorem
313(2)
14.6 Euler's recursion formula for ρ(n)
315(1)
14.7 An upper bound for ρ(n)
316(2)
14.8 Jacobi's triple product identity
318(3)
14.9 Consequences of Jacobi's identity
321(1)
14.10 Logarithmic differentiation of generating functions
322(2)
14.11 The partition identities of Ramanujan
324(5)
Exercises for
Chapter 14
325(4)
Bibliography 329(4)
Index of Special Symbols 333(2)
Index 335
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