Abstract analytic number theory

1st Edition, Volume 12 - January 1, 1975
Editor: Knopfmacher
Language: English
eBook ISBN:
9 7 8 - 0 - 4 4 4 - 1 0 7 7 9 - 4

North-Holland Mathematical Library, Volume 12: Abstract Analytic Number Theory focuses on the approaches, methodologies, and principles of the abstract analytic number theory.… Read more

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North-Holland Mathematical Library, Volume 12: Abstract Analytic Number Theory focuses on the approaches, methodologies, and principles of the abstract analytic number theory. The publication first deals with arithmetical semigroups, arithmetical functions, and enumeration problems. Discussions focus on special functions and additive arithmetical semigroups, enumeration and zeta functions in special cases, infinite sums and products, double series and products, integral domains and arithmetical semigroups, and categories satisfying theorems of the Krull-Schmidt type. The text then ponders on semigroups satisfying Axiom A, asymptotic enumeration and "statistical" properties of arithmetical functions, and abstract prime number theorem. Topics include asymptotic properties of prime-divisor functions, maximum and minimum orders of magnitude of certain functions, asymptotic enumeration in certain categories, distribution functions of prime-independent functions, and approximate average values of special arithmetical functions. The manuscript takes a look at arithmetical formations, additive arithmetical semigroups, and Fourier analysis of arithmetical functions, including Fourier theory of almost even functions, additive abstract prime number theorem, asymptotic average values and densities, and average values of arithmetical functions over a class. The book is a vital reference for researchers interested in the abstract analytic number theory.

Preface

Introduction

Part I. Arithmetical Semigroups and Algebraic Enumeration Problems

Chapter 1. Arithmetical Semigroups

§1. Integral Domains and Arithmetical Semigroups

§2. Categories Satisfying Theorems of the Krull-Schmidt Type

Chapter 2. Arithmetical Functions

§1. The Dirichlet Algebra of an Arithmetical Semigroup

§2. Infinite Sums and Products

§3. Double Series and Products

§4. Types of Arithmetical Functions

§5. The Zeta and Möbius Functions

§6. Further Natural Arithmetical Functions

§7. ζ-Formulae

Chapter 3. Enumeration Problems

§1. A Special Algebra Homomorphism

§2. Enumeration and Zeta Functions in Special Cases

§3. Special Functions and Additive Arithmetical Semigroups

Part II. Arithmetical Semigroups with Analytical Properties of Classical Type

Chapter 4. Semigroups Satisfying Axiom A

§1. The Basic Axiom

§2. Analytical Properties of the Zeta Function

§3. Average Values of Arithmetical Functions

§4. Approximate Average Values of Special Arithmetical Functions

§5. Asymptotic Formulae with Error Estimates

Chapter 5. Asymptotic Enumeration, and Further “Statistical” Properties of Arithmetical Functions

§1. Asymptotic Enumeration in Certain Categories

§2. Maximum Orders of Magnitude

§3. Distribution Functions of Prime-Independent Functions

Chapter 6. The Abstract Prime Number Theorem

§1. The Fundamental Theorem

§2. Asymptotic Properties of Prime-Divisor Functions

§3. Maximum and Minimum Orders of Magnitude of Certain Functions

§4. The “Law of Large Numbers” for Certain Functions

Chapter 7. Fourier Analysis of Arithmetical Functions

§1. Algebraic and Topological Theory of Ramanujan Sums

§2. Fourier Theory of Even Functions

§3. Fourier Theory of Almost Even Functions

§4. A Wider Type of Almost Evenness, and Pointwise Convergence of Ramanujan Expansions

§5. Arithmetical Functions Over Gz

Part III. Analytical Properties of Other Arithmetical Systems

Chapter 8. Additive Arithmetical Semigroups

§1. Axiom C

§2. Analytical Properties of the Zeta Function

§3. The Additive Abstract Prime Number Theorem

§4. Further Additive Prime Number Theorems

§5. Asymptotic Average Values and Densities

Chapter 9. Arithmetical Formations

§1. Natural Examples

§2. Characters and Formations

§3. The L-Series of a Formation

§4. Axiom A*

§5. Analytical Properties of L-Series

§6. Average Values of Arithmetical Functions Over a Class

§7. Abstract Prime Number Theorem for Formations

Appendix 1. Some Unsolved Questions

Appendix 2. Values of p(n) and s(n)

List of Special Symbols

Bibliography

Index