Number Theory Arising From Finite Fields: Analytic And Probabilistic Theory

by Knopfmacher; John

• ISBN13: 9780824705770
• ISBN10: 0824705777
• Format: Hardcover
• Copyright: 2001-04-10
• Publisher: CRC Press

Summary

Reference text providing coverage of the latest advances in the field of number theory, with an emphasis on abstract prime number theorems, mean-value theorems of multiplicative functions, and the normal distribution of additive factors.

Table of Contents

Preface	iii
Introduction	1	(4)
Additive Arithmetical Semigroups and Axiom A#	5	(44)
Basic Concepts and Examples	5	(12)
The Zeta or Generating Function	17	(8)
Averages and Densities in Simple Cases	25	(7)
Asymptotic Moments of Specific Functions	32	(6)
Error Estimates for Certain Averages	38	(11)
Asymptotic Enumeration and More Refined Estimates	49	(22)
Asymptotic Enumeration of Modules and Algebras	49	(7)
Sharper Average and Enumerative Estimates	56	(15)
Abstract Prime Number Theorems (I)	71	(92)
General Remarks and Preliminary Results	71	(17)
Chebyshev-Type Upper Bounds	88	(9)
Mertens-Type Asymptotic Estimates and Prime Divisor Functions	97	(17)
Abstract Prime Number Theorems and Zeros of the Generating Function	114	(6)
Explicit Abstract Prime Number Theorems	120	(9)
A Tauberian Theorem of Bombieri	129	(11)
Elementary Proofs	140	(10)
Two Analytical Examples	150	(13)
More Applications of Prime Counting	163	(26)
More Properties of Prime Divisor Functions in the ``Classical'' Case	163	(17)
Maximum Orders of Magnitude in the ``Classical'' Case	180	(9)
Abstract Prime Number Theorems (II)	189	(38)
A Theorem of Indlekofer-Manstavicius-Warlimont	189	(5)
The Total Number of Zeros of the Generating Function	194	(10)
The Orders of the Zeros	204	(7)
An Alternative Abstract Prime Number Theorem	211	(7)
Beurling-Type Abstract Prime Number Theorems	218	(9)
General Mean-Value Theorems	227	(55)
Preliminaries	227	(15)
General Mean-Value Theorems	242	(10)
An Analogue of the Halasz-Wirsing Theorem	252	(15)
Mean-Values of Multiplicative Functions	267	(7)
Mean-Values of the Functions λ and μ	274	(5)
Mean-Value Theorems for Beurling-Type Semigroups	279	(3)
Basics of Probabilistic Number Theory for Semigroups	282	(71)
Necessary Results from Theory of Probability	282	(15)
Limit Distribution Functions of Real-Valued Additive Functions	297	(6)
An Analogue of the Turan-Kubilius Inequality	303	(12)
A Fundamental Lemma	315	(14)
A Probability Model	329	(8)
Infinitely Divisible Distributions and a Central Limit Theorem	337	(14)
Rate of Convergence to the Normal Law	351	(2)
Survey of Some Further Topics	353	(32)
Asymptotics of Factorizations	353	(6)
Direct Factors of Arithmetical Semigroups	359	(2)
Sets of Multiples	361	(3)
Further Properties of Special Sets and Functions	364	(3)
More About Polynomials and Finite Fields	367	(6)
Ramanujan Expansions of Arithmetical Functions	373	(5)
Additive Arithmetical Formations	378	(7)
Bibliography	385	(12)
Notations	397	(4)
Index	401