Introduction to Analytic Number Theory

Автор(ы):Tom M. Apostol
06.10.2007
Год изд.:1976
Описание: This is the first volume of a two-volume textbook which evolved from a course (Mathematics 160) offered at the California Institute of Technology during the last 25 years. It provides an introduction to analytic number theory suitable for undergraduates with some background in advanced calculus, but with no previous knowledge of number theory. Actually, a great deal of the book requires no calculus at all and could profitably be studied by sophisticated high school students. Number theory is such a vast and rich field that a one-year course cannot do justice to all its parts. The choice of topics included here is intended to provide some variety and some depth. Problems which have fascinated generations of professional and amateur mathematicians are discussed together with some of the techniques for solving them. One of the goals of this course has been to nurture the intrinsic interest that many young mathematics students seem to have in number theory and to open some doors for them to the current periodical literature. It has been gratifying to note that many of the students who have taken this course during the past 25 years have become professional mathematicians, and some have made notable contributions of their own to number theory. To all of them this book is dedicated.
Оглавление:
Introduction to Analytic Number Theory — обложка книги.
Contents
Historical Introduction
Chapter 1
The Fundamental Theorem of Arithmetic
  1.1 Introduction [13]
  1.2 Divisibility [14]
  1.3 Greatest common divisor [14]
  1.4 Prime numbers [16]
  1.5 The fundamental theorem of arithmetic [17]
  1.6 The series of reciprocals of the primes [18]
  1.7 The Euclidean algorithm [19]
  1.8 The greatest common divisor of more than two numbers [20]
      Exercises for Chapter 1 [21]
Chapter 2
Arithmetical Functions and Dirichlet Multiplication
  2.1 Introduction [24]
  2.2 The Mobius function (?)(?) [24]
  2.3 The Euler totient function (?)(?) [25]
  2.4 A relation connecting (?) and (?) [26]
  2.5 A product formula for (?)(?) [27]
  2.6 The Dirichlet product of arithmetical functions [29]
  2.7 Dirichlet inverses and the Mobius inversion formula [30]
  2.8 The Mangoldt function (?)(?) [32]
  2.9 Multiplicative functions [33]
  2.10 Multiplicative functions and Dirichlet multiplication [35]
  2.11 The inverse of a completely multiplicative function [36]
  2.12 Liouville's function (?)(?) [37]
  2.13 The divisor functions (?)(?) [38]
  2.14 Generalized convolutions [39]
  2.15 Formal power series [41]
  2.16 The Bell series of an arithmetical function [42]
  2.17 Bell series and Dirichlet multiplication [44]
  2.18 Derivatives of arithmetical functions [45]
  2.19 The Selberg identity [46]
       Exercises for Chapter 2 [46]
Chapter 3
Averages of Arithmetical Functions
  3.1 Introduction [52]
  3.2 The big oh notation. Asymptotic equality of functions [53]
  3.3 Euler's summation formula [54]
  3.4 Some elementary asymptotic formulas [55]
  3.5 The average order of (?)(?) [57]
  3.6 The average order of the divisor functions (?)(?) [60]
  3.7 The average order of (?)(?) [61]
  3.8 An application to the distribution of lattice points visible from the origin [62]
  3.9 The average order of ц(п) and of (?)(?) [64]
  3.10 The partial sums of a Dirichlet product [65]
  3.11 Applications to (?)(?) and (?)(?) 66]
  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 [74]
  4.2 Chebyshev's functions (?)(?) and (?)(?) [75]
  4.3 Relations connecting (?)(?) and (?)(?) [76]
  4.4 Some equivalent forms of the prime number theorem [79]
  4.5 Inequalities for (?)(?) and (?)(?) [82]
  4.6 Shapiro's Tauberian theorem [85]
  4.7 Applications of Shapiro's theorem [88]
  4.8 An asymptotic formula for the partial sums (формула) [89]
  4.9 The partial sums of the Mobius function [91]
  4.10 Brief sketch of an elementary proof of the prime number theorem [98]
  4.11 Selberg's asymptotic formula [99]
       Exercises for Chapter 4 [101]
Chapter 5 Congruences
  5.1 Definition and basic properties of congruences [106]
  5.2 Residue classes and complete residue systems [109]
  5.3 Linear congruences [110]
  5.4 Reduced residue systems and the Euler-Fermat theorem [113]
  5.5 Polynomial congruences modulo p. Lagrange's theorem [114]
  5.6 Applications of Lagrange's theorem [115]
  5.7 Simultaneous linear congruences. The Chinese remainder theorem [117]
  5.8 Applications of the Chinese remainder theorem [118]
  5.9 Polynomial congruences with prime power moduli [120]
  5.10 The principle of cross-classification [123]
  5.11 A decomposition property of reduced residue systems [125]
       Exercises for Chapter 5 [126]
Chapter 6
Finite Abelian Groups and Their Characters
  6.1 Definitions [129]
  6.2 Examples of groups and subgroups [130]
  6.3 Elementary properties of groups [130]
  6.4 Construction of subgroups [131]
  6.5 Characters of finite abelian groups [133]
  6.6 The character group [135]
  6.7 The orthogonality relations for characters [136]
  6.8 Dirichlet characters [137]
  6.9 Sums involving Dirichlet characters [140]
  6.10 The nonvanishing of L(?)(?) for real nonprincipal (?) [141]
       Exercises for Chapter 6 [143]
Chapter 7
Dirichlet's Theorem on Primes in Arithmetic Progressions
  7.1 Introduction [146]
  7.2 Dirichlet's theorem for primes of the form 4n — 1 and 4n + 1 [147]
  7.3 The plan of the proof of Dirichlet's theorem [148]
  7.4 Proof of Lemma 7.4 [150]
  7.5 Proof of Lemma 7.5 [151]
  7.6 Proof of Lemma 7.6 [152]
  7.7 Proof of Lemma 7.8 [153]
  7.8 Proof of Lemma 7.7 [153]
  7.9 Distribution of primes in arithmetic progressions [154]
      Exercises for Chapter 7 [155]
Chapter 8
Periodic Arithmetical Functions and Gauss Sums
  8.1 Functions periodic modulo (?) [157]
  8.2 Existence of finite Fourier series for periodic arithmetical functions [158]
  8.3 Ramanujan's sum and generalizations [160]
  8.4 Multiplicative properties of the sums (?)(?) [162]
  8.5 Gauss sums associated with Dirichlet characters [165]
  8.6 Dirichlet characters with nonvanishing Gauss sums [166]
  8.7 Induced moduli and primitive characters [167]
  8.8 Further properties of induced moduli [168]
  8.9 The conductor of a character [171]
  8.10 Primitive characters and separable Gauss sums [171]
  8.11 The finite Fourier series of the Dirichlet characters [172]
  8.12 Polya's inequality for the partial sums of primitive characters [173]
       Exercises for Chapter 8 [175]
Chapter 9
Quadratic Residues and the Quadratic Reciprocity Law
  9.1 Quadratic residues [178]
  9.2 Legendre's symbol and its properties [179]
  9.3 Evaluation of (-1 |p) and (2|p) [181]
  9.4 Gauss' lemma [182]
  9.5 The quadratic reciprocity law [185]
  9.6 Applications of the reciprocity law [186]
  9.7 The Jacobi symbol [187]
  9.8 Applications to Diophantine equations [190]
  9.9 Gauss sums and the quadratic reciprocity law [192]
  9.10 The reciprocity law for quadratic Gauss sums [195]
  9.11 Another proof of the quadratic reciprocity law 200 Exercises for Chapter 9 [201]
Chapter 10 Primitive Roots
  10.1 The exponent of a number mod m. Primitive roots [204]
  10.2 Primitive roots and reduced residue systems [205]
  10.3 The nonexistence of primitive roots mod (?) for (?)(?) [206]
  10.4 The existence of primitive roots mod (?) for odd primes (?) [206]
  10.5 Primitive roots and quadratic residues [208]
  10.6 The existence of primitive roots mod (?) [208]
  10.7 The existence of primitive roots mod (?) [210]
  10.8 The nonexistence of primitive roots in the remaining cases [211]
  10.9 The number of primitive roots mod (?) [212]
  10.10 The index calculus [273]
  10.11 Primitive roots and Dirichlet characters [278]
  10.12 Real-valued Dirichlet characters mod (?) [220]
  10.13 Primitive Dirichlet characters mod (?) [221]
        Exercises for Chapter 10 [222]
Chapter 11
Dirichlet Series and Euler Products
  11.1 Introduction [224]
  11.2 The half-plane of absolute convergence of a Dirichlet series [225]
  11.3 The function defined by a Dirichlet series [226]
  11.4 Multiplication of Dirichlet series [228]
  11.5 Euler products [230]
  11.6 The half-plane of convergence of a Dirichlet series [232]
  11.7 Analytic properties of Dirichlet series [234]
  11.8 Dirichlet series with nonnegative coefficients [236]
  11.9 Dirichlet series expressed as exponentials of Dirichlet series [238]
  11.10 Mean value formulas for Dirichlet series [240]
  11.11 An integral formula for the coefficients of a Dirichlet series [242]
  11.12 An integral formula for the partial sums of a Dirichlet series [243]
        Exercises for Chapter 11 [246]
Chapter 12
The Functions (?)(?) and (?)(?)
  12.1 Introduction [249]
  12.2 Properties of the gamma function [250]
  12.3 Integral representation for the Hurwitz zeta function [251]
  12.4 A contour integral representation for the Hurwitz zeta function [253]
  12.5 The analytic continuation of the Hurwitz zeta function [254]
  12.6 Analytic continuation of (?)(?) and (?)(?) [255]
  12.7 Hurwitz's formula for (?)(?) [256]
  12.8 The functional equation for the Riemann zeta function [259]
  12.9 A functional equation for the Hurwitz zeta function [260]
  12.10 The functional equation for L-functions [261]
  12.11 Evaluation of (?)(?) [264]
  12.12 Properties of Bernoulli numbers and Bernoulli polynomials [265]
  12.13 Formulas for L(0, x) [268]
  12.14 Approximation of (?)(?) by finite sums [268]
  12.15 Inequalities for |(?)(?) [270]
  12.16 Inequalities for (?)(?) and (?)(?) [272]
        Exercises for Chapter 12 [273]
Chapter 13
Analytic Proof of the Prime Number Theorem
  13.1 The plan of the proof [278]
  13.2 Lemmas [279]
  13.3 A contour integral representation for (?)(?) [283]
  13.4 Upper bounds for (?)(?) and (?)(?) near the line a (?) [284]
  13.5 The nonvanishing of (?) on the line (?) [286]
  13.6 Inequalities for (?)(?) and (?)(?) [287]
  13.7 Completion of the proof of the prime number theorem [289]
  13.8 Zero-free regions for (?) [291]
  13.9 The Riemann hypothesis [293]
  13.10 Application to the divisor function [294]
  13.11 Application to Euler's totient [297]
  13.12 Extension of Polya's inequality for character sums [299]
        Exercises for Chapter 13 [300]
Chapter 14 Partitions
  14.1 Introduction [304]
  14.2 Geometric representation of partitions [307]
  14.3 Generating functions for partitions [308]
  14.4 Euler's pentagonal-number theorem [311]
  14.5 Combinatorial proof of Euler's pentagonal-number theorem [313]
  14.6 Euler's recursion formula for (?) [315]
  14.7 An upper bound for (?) [316]
  14.8 Jacobi's triple product identity [318]
  14.9 Consequences of Jacobi's identity [321]
  14.10 Logarithmic differentiation of generating functions [322]
  14.11 The partition identities of Ramanujan [324]
        Exercises for Chapter 14 [325]
Bibliography [329]
Index of Special Symbols [333]
Index [335]
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