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Complex analysis in number theory

Complex analysis in number theory (1회 대출)

자료유형
단행본
개인저자
Karatsuba, Anatoly Alekseevich.
서명 / 저자사항
Complex analysis in number theory / Anatoly A. Karatsuba.
발행사항
Boca Raton :   CRC Press,   c1995.  
형태사항
ix, 187 p. : ill. ; 25 cm.
ISBN
0849328667 (acid-free paper)
서지주기
Includes bibliographical references (p. 171-182) and index.
일반주제명
Number theory. Functions of complex variables. Mathematical analysis. Nombres, theorie des. Fonctions d'une variable complexe. Analyse mathematique.
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100 1 ▼a Karatsuba, Anatoly Alekseevich.
245 1 0 ▼a Complex analysis in number theory / ▼c Anatoly A. Karatsuba.
260 ▼a Boca Raton : ▼b CRC Press, ▼c c1995.
300 ▼a ix, 187 p. : ▼b ill. ; ▼c 25 cm.
504 ▼a Includes bibliographical references (p. 171-182) and index.
650 0 ▼a Number theory.
650 0 ▼a Functions of complex variables.
650 0 ▼a Mathematical analysis.
650 7 ▼a Nombres, theorie des. ▼2 ram
650 7 ▼a Fonctions d'une variable complexe. ▼2 ram
650 7 ▼a Analyse mathematique. ▼2 ram

소장정보

No. 소장처 청구기호 등록번호 도서상태 반납예정일 예약 서비스
No. 1 소장처 중앙도서관/서고7층/ 청구기호 512.73 K18c 등록번호 111049987 도서상태 대출가능 반납예정일 예약 서비스 B M
No. 2 소장처 중앙도서관/서고7층/ 청구기호 512.73 K18c 등록번호 111083894 도서상태 대출가능 반납예정일 예약 서비스 B M

컨텐츠정보

목차


CONTENTS
Introduction = 1
Chapter 1. The Complex Integration Method and Its Application in Number Theory = 8
 1. Generating Functions in Number Theory = 8
  1.1 Dirichlet's series = 8
  1.2 Sum functions = 11
 2. Summation Formula = 13
  2.1 Perron's formula = 13
  2.2 Expressing Chebyshev's function in terms of the integral of the logarithnlic derivative of Riemann's zeta-function = 14
 3. Riemann's Zeta-Function and Its Simplest Properties = 15
  3.1 The functional equation = 15
  3.2 Riemann's hypotheses = 17
  3.3 The simplest theorems on the zeros of ζ(s) = 18
  3.4 Expressing Chebyshev's function as a sum over the complex zeros of ζ(s) = 19
  3.5 The asymptotic law of distribution of prime numbErs = 20
  3.6 Riemaiin's hypothesis concerning the complex zeros of ζ(s) and the problem of the theory of prime numbers = 21
  3.7 Theorem on the uniqueness of ζ(s) = 23
  3.8 Proofs of the simplest theorems on the complex zeros of ζ(s) = 24
Chapter 2. The Theory of Riemann's Zeta-Function = 31
 1. Zeros on the Critical Line = 31
  1.1 Hardy's theorem = 31
  1.2 Theorems of Hardy and Littlewood = 31
  1.3 Hardy's function and Hardy's method = 32
  1.4 Titchmarsh's discrete method = 35
  1.5 Selberg's theorem = 35
  1.6 Estimates of Selberg's constant = 36
  1.7 Moser's theorems = 37
  1.8 Selberg's hypothesis = 38
  1.9 Zeros of the derivatives of Hardy's function = 39
  1.10 The latest results = 40
  1.11 Distribution of zeros in the mean = 41
  1.12 Density of zeros on the critical line = 41
  1.13 The zeros of ζ(s) in the neighborhood of the critical line = 42
 2. The Boundary of Zeros = 43
  2.1 De la Valke Poussin theorem = 43
  2.2 Littlewood's theorem = 43
  2.3 The relationship between the boundary of zeros and the order of growth of │ζ(s)│in the neighborhood of unit line = 44
  2.4 Vinogradov's method in the theory of ζ(s) and Chudakov's theorems = 45
  2.5 Vinogradov's theorem = 46
 3. Approximate Equations of the ζ(s) Function = 47
  3.1 Partial summation and Euler's summation formula = 47
  3.2 The simplest approximation of ζ(s) = 49
  3.3 The approximation of a trigonometric sum by a sum of trigonometric integrals = 50
  3.4 Asymptotic calculations of a certain dass of trigonometric integrals = 57
  3.5 Approximation of a trigonometric sum by a more concise sum = 66
  3.6 Approximate equations of the ζ(s) function = 69
  3.7 On trigonometric integrals = 73
 4. The Method of Trigonometric Sums in the Theory of the ζ(s) Function = 77
  4.1 The meaa value. of the degree of the modulus of a trigonometric sum = 77
  4.2 Simple lemmas = 78
  4.3 The basic recurrent inequality = 83
  4.4 Vinogradov's mean-value theorem = 89
  4.5 The estimate of the zeta sum and its consequences = 91
  4.6 The current boundary of zeros of ζ(s) and its corollaries = 98
 5. Density TheoTems = 100
  5.1 Bertrand's postulate and Chebyshev's theorem = 100
  5.2 Hoheisel's method = 100
  5.3 Density of zeros of ζ(s) = 102
  5.4 Density theorems = 103
  5.5 Proof of Huxley's density theorem = 104
  5.6 Three problems of the number theory solvable by Hoheisel's method = 120
 6. The Order of Growth of │ζ(s)│ in a Critical Strip = 122
  6.1 The problem of Dirichlet's di'visors = 123
  6.2 Lindel6f's hypothesis = 124
  6.3 Equivalents of Lindel6f's hypothesis = 125
  6.4 The order of growth of │ζ($${1 \over 2}$$+ it )│= 126
  6.5 Vinogradov's method, in Dirichlet's multi-dimensional divisor problem = 127
  6.6 Omega-theorems = 130
 7. Universal Properties of the ζ(s) Function = 130
  7.1 Bohr's theorems = 130
  7.2 Voronin's theorems = 132
  7.3 Theorem on the universal character of ζ(s) = 134
  7.4 More on the universal character of ζ(s) = 135
 8. Riemann's Hypothesis, Its Equivalents, Computations = 135
  8.1 Mertens' hypothesis = 136
  8.2 Turan's hypothesis and its refutation = 137
  8.3 A billion and a half complex zeros of ζ(s) = 138
  8.4 Computations connected with ζ(s) = 138
  8.5 Functions resembling ζ(s) but having complex zeros on the right of the critical line = 139
  8.6 Epstein's zeta-functions = 140
  8.7 A new approach to the problem of zeros, lying on the critical line, of some Dirichlet series = 141
Chapter 3. Dirichlet L-Functions = 147
 1. Dirichlet's Characters = 147
  1.1 Definition of character's = 147
  1.2 Principal properties of characters = 148
 2. Dirichlet L-Functions and Prime Numbers in Arithmetic Progressions = 149
  2.1 Definition of L-functions = 149
  2.2 The functions π(x; k, 1) and ψ(x; k, 1) = 150
  2.3 Dirichlet's theorem on primes = 150
 3. Zeros of L-Functions = 152
  3.1 The boundary of zeros. Page's theorems = 152
  3.2 Siegel's theorem = 153
  3.3 Zeros on the critical line = 153
 4. Real Zeros of L-Functions and the Number of Classes of Binary Quadratic Forms = 154
  4.1 Binary quadratic forms and the number of dasses = 154
  4.2 Dirichlet's formulas = 156
  4.3 Gauss' problem and Siegel's theorem = 156
  4.4 Prime numbers in arithmetic progressions = 157
 5. Density Theorems = 158
  5.1 Linnik's density theorems = 158
  5.2 Density theorems of a large sieve and the Bombieri-Vinogradov theorem = 158
  5.3 Current density theorems = 160
  5.4 Proof of Vinogradov's theorem on three prime numbers based on the ideas of Hardy-Littlewood-Linnik = 160
 6. L-Functions and Nonresidues = 163
  6.1 The concept of a nonresidue = 163
  6.2 Vinogradov's hypothesis = 163
  6.3 Lindel$$\ddot o$$f's generalized hypothesis and a nonresidue = 164
  6.4 The zeros of the L-function: and nonresidues = 164
 7. Approximate Equations = 165
  7.1 Stating the problem = 165
  7.2 Lavrik's general theorem = 165
 8. On Primitive Roots = 168
  8.1 The concept of a primitive root = 168
  8.2 Artin's hypothesis = 168
  8.3 Hooley's conditional theorem = 168
References = 170
Author Index = 183
Subject Index = 185


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