 Introduction to Analytic Number Theory (Tom M. Apostol)

# 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: Обложка книги. Contents Historical Introduction Chapter 1 The Fundamental Theorem of Arithmetic   1.1 Introduction    1.2 Divisibility    1.3 Greatest common divisor    1.4 Prime numbers    1.5 The fundamental theorem of arithmetic    1.6 The series of reciprocals of the primes    1.7 The Euclidean algorithm    1.8 The greatest common divisor of more than two numbers        Exercises for Chapter 1  Chapter 2 Arithmetical Functions and Dirichlet Multiplication   2.1 Introduction    2.2 The Mobius function (?)(?)    2.3 The Euler totient function (?)(?)    2.4 A relation connecting (?) and (?)    2.5 A product formula for (?)(?)    2.6 The Dirichlet product of arithmetical functions    2.7 Dirichlet inverses and the Mobius inversion formula    2.8 The Mangoldt function (?)(?)    2.9 Multiplicative functions    2.10 Multiplicative functions and Dirichlet multiplication    2.11 The inverse of a completely multiplicative function    2.12 Liouville's function (?)(?)    2.13 The divisor functions (?)(?)    2.14 Generalized convolutions    2.15 Formal power series    2.16 The Bell series of an arithmetical function    2.17 Bell series and Dirichlet multiplication    2.18 Derivatives of arithmetical functions    2.19 The Selberg identity         Exercises for Chapter 2  Chapter 3 Averages of Arithmetical Functions   3.1 Introduction    3.2 The big oh notation. Asymptotic equality of functions    3.3 Euler's summation formula    3.4 Some elementary asymptotic formulas    3.5 The average order of (?)(?)    3.6 The average order of the divisor functions (?)(?)    3.7 The average order of (?)(?)    3.8 An application to the distribution of lattice points visible from the origin    3.9 The average order of ц(п) and of (?)(?)    3.10 The partial sums of a Dirichlet product    3.11 Applications to (?)(?) and (?)(?) 66]   3.12 Another identity for the partial sums of a Dirichlet product         Exercises for Chapter 3  Chapter 4 Some Elementary Theorems on the Distribution of Prime Numbers   4.1 Introduction    4.2 Chebyshev's functions (?)(?) and (?)(?)    4.3 Relations connecting (?)(?) and (?)(?)    4.4 Some equivalent forms of the prime number theorem    4.5 Inequalities for (?)(?) and (?)(?)    4.6 Shapiro's Tauberian theorem    4.7 Applications of Shapiro's theorem    4.8 An asymptotic formula for the partial sums (формула)    4.9 The partial sums of the Mobius function    4.10 Brief sketch of an elementary proof of the prime number theorem    4.11 Selberg's asymptotic formula         Exercises for Chapter 4  Chapter 5 Congruences   5.1 Definition and basic properties of congruences    5.2 Residue classes and complete residue systems    5.3 Linear congruences    5.4 Reduced residue systems and the Euler-Fermat theorem    5.5 Polynomial congruences modulo p. Lagrange's theorem    5.6 Applications of Lagrange's theorem    5.7 Simultaneous linear congruences. The Chinese remainder theorem    5.8 Applications of the Chinese remainder theorem    5.9 Polynomial congruences with prime power moduli    5.10 The principle of cross-classification    5.11 A decomposition property of reduced residue systems         Exercises for Chapter 5  Chapter 6 Finite Abelian Groups and Their Characters   6.1 Definitions    6.2 Examples of groups and subgroups    6.3 Elementary properties of groups    6.4 Construction of subgroups    6.5 Characters of finite abelian groups    6.6 The character group    6.7 The orthogonality relations for characters    6.8 Dirichlet characters    6.9 Sums involving Dirichlet characters    6.10 The nonvanishing of L(?)(?) for real nonprincipal (?)         Exercises for Chapter 6  Chapter 7 Dirichlet's Theorem on Primes in Arithmetic Progressions   7.1 Introduction    7.2 Dirichlet's theorem for primes of the form 4n — 1 and 4n + 1    7.3 The plan of the proof of Dirichlet's theorem    7.4 Proof of Lemma 7.4    7.5 Proof of Lemma 7.5    7.6 Proof of Lemma 7.6    7.7 Proof of Lemma 7.8    7.8 Proof of Lemma 7.7    7.9 Distribution of primes in arithmetic progressions        Exercises for Chapter 7  Chapter 8 Periodic Arithmetical Functions and Gauss Sums   8.1 Functions periodic modulo (?)    8.2 Existence of finite Fourier series for periodic arithmetical functions    8.3 Ramanujan's sum and generalizations    8.4 Multiplicative properties of the sums (?)(?)    8.5 Gauss sums associated with Dirichlet characters    8.6 Dirichlet characters with nonvanishing Gauss sums    8.7 Induced moduli and primitive characters    8.8 Further properties of induced moduli    8.9 The conductor of a character    8.10 Primitive characters and separable Gauss sums    8.11 The finite Fourier series of the Dirichlet characters    8.12 Polya's inequality for the partial sums of primitive characters         Exercises for Chapter 8  Chapter 9 Quadratic Residues and the Quadratic Reciprocity Law   9.1 Quadratic residues    9.2 Legendre's symbol and its properties    9.3 Evaluation of (-1 |p) and (2|p)    9.4 Gauss' lemma    9.5 The quadratic reciprocity law    9.6 Applications of the reciprocity law    9.7 The Jacobi symbol    9.8 Applications to Diophantine equations    9.9 Gauss sums and the quadratic reciprocity law    9.10 The reciprocity law for quadratic Gauss sums    9.11 Another proof of the quadratic reciprocity law 200 Exercises for Chapter 9  Chapter 10 Primitive Roots   10.1 The exponent of a number mod m. Primitive roots    10.2 Primitive roots and reduced residue systems    10.3 The nonexistence of primitive roots mod (?) for (?)(?)    10.4 The existence of primitive roots mod (?) for odd primes (?)    10.5 Primitive roots and quadratic residues    10.6 The existence of primitive roots mod (?)    10.7 The existence of primitive roots mod (?)    10.8 The nonexistence of primitive roots in the remaining cases    10.9 The number of primitive roots mod (?)    10.10 The index calculus    10.11 Primitive roots and Dirichlet characters    10.12 Real-valued Dirichlet characters mod (?)    10.13 Primitive Dirichlet characters mod (?)          Exercises for Chapter 10  Chapter 11 Dirichlet Series and Euler Products   11.1 Introduction    11.2 The half-plane of absolute convergence of a Dirichlet series    11.3 The function defined by a Dirichlet series    11.4 Multiplication of Dirichlet series    11.5 Euler products    11.6 The half-plane of convergence of a Dirichlet series    11.7 Analytic properties of Dirichlet series    11.8 Dirichlet series with nonnegative coefficients    11.9 Dirichlet series expressed as exponentials of Dirichlet series    11.10 Mean value formulas for Dirichlet series    11.11 An integral formula for the coefficients of a Dirichlet series    11.12 An integral formula for the partial sums of a Dirichlet series          Exercises for Chapter 11  Chapter 12 The Functions (?)(?) and (?)(?)   12.1 Introduction    12.2 Properties of the gamma function    12.3 Integral representation for the Hurwitz zeta function    12.4 A contour integral representation for the Hurwitz zeta function    12.5 The analytic continuation of the Hurwitz zeta function    12.6 Analytic continuation of (?)(?) and (?)(?)    12.7 Hurwitz's formula for (?)(?)    12.8 The functional equation for the Riemann zeta function    12.9 A functional equation for the Hurwitz zeta function    12.10 The functional equation for L-functions    12.11 Evaluation of (?)(?)    12.12 Properties of Bernoulli numbers and Bernoulli polynomials    12.13 Formulas for L(0, x)    12.14 Approximation of (?)(?) by finite sums    12.15 Inequalities for |(?)(?)    12.16 Inequalities for (?)(?) and (?)(?)          Exercises for Chapter 12  Chapter 13 Analytic Proof of the Prime Number Theorem   13.1 The plan of the proof    13.2 Lemmas    13.3 A contour integral representation for (?)(?)    13.4 Upper bounds for (?)(?) and (?)(?) near the line a (?)    13.5 The nonvanishing of (?) on the line (?)    13.6 Inequalities for (?)(?) and (?)(?)    13.7 Completion of the proof of the prime number theorem    13.8 Zero-free regions for (?)    13.9 The Riemann hypothesis    13.10 Application to the divisor function    13.11 Application to Euler's totient    13.12 Extension of Polya's inequality for character sums          Exercises for Chapter 13  Chapter 14 Partitions   14.1 Introduction    14.2 Geometric representation of partitions    14.3 Generating functions for partitions    14.4 Euler's pentagonal-number theorem    14.5 Combinatorial proof of Euler's pentagonal-number theorem    14.6 Euler's recursion formula for (?)    14.7 An upper bound for (?)    14.8 Jacobi's triple product identity    14.9 Consequences of Jacobi's identity    14.10 Logarithmic differentiation of generating functions    14.11 The partition identities of Ramanujan          Exercises for Chapter 14  Bibliography  Index of Special Symbols  Index  Формат: djvu Размер: 2780984 байт Язык: ENG Рейтинг: 467 Открыть: Ссылка (RU)