Hardcover ISBN:  9781470430450 
Product Code:  SURV/213 
244 pp 
List Price:  $110.00 
MAA Member Price:  $99.00 
AMS Member Price:  $88.00 
Electronic ISBN:  9781470435387 
Product Code:  SURV/213.E 
244 pp 
List Price:  $110.00 
MAA Member Price:  $99.00 
AMS Member Price:  $88.00 

Book DetailsMathematical Surveys and MonographsVolume: 213; 2016MSC: Primary 11;
“Generalized numbers” is a multiplicative structure introduced by A. Beurling to study how independent prime number theory is from the additivity of the natural numbers. The results and techniques of this theory apply to other systems having the character of prime numbers and integers; for example, it is used in the study of the prime number theorem (PNT) for ideals of algebraic number fields.
Using both analytic and elementary methods, this book presents many old and new theorems, including several of the authors' results, and many examples of extremal behavior of gnumber systems. Also, the authors give detailed accounts of the \(L^2\) PNT theorem of J. P. Kahane and of the example created with H. L. Montgomery, showing that additive structure is needed for proving the Riemann hypothesis.
Other interesting topics discussed are propositions “equivalent” to the PNT, the role of multiplicative convolution and Chebyshev's prime number formula for gnumbers, and how Beurling theory provides an interpretation of the smooth number formulas of Dickman and de Bruijn.ReadershipGraduate students and researchers interested in analytic number theory.

Table of Contents

Chapters

Chapter 1. Overview

Chapter 2. Analytic machinery

Chapter 3. $\mathbf {dN}$ as an exponential and Chebyshev’s identity

Chapter 4. Upper and lower estimates of $\mathbf {N(x)}$

Chapter 5. Mertens’ formulas and logarithmic density

Chapter 6. Odensity of gintegers

Chapter 7. Density of gintegers

Chapter 8. Simple estimates of $\mathbf {\pi (x)}$

Chapter 9. Chebyshev bounds—Elementary theory

Chapter 10. WienerIkehara Tauberian theorems

Chapter 11. Chebyshev bounds—Analytic methods

Chapter 12. Optimality of a Chebyshev bound

Chapter 13. Beurling’s PNT

Chapter 14. Equivalences to the PNT

Chapter 15. Kahane’s PNT

Chapter 16. PNT with remainder

Chapter 17. Optimality of the dlVP remainder term

Chapter 18. The Dickman and Buchstab functions


Additional Material

Reviews

The book is very well written and should prove accessible to a fairly large audience, requiring only some familiarity with mathematical analysis and classical techniques from analytic number theory.
JeanMarie De Koninck, Mathematical Reviews


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“Generalized numbers” is a multiplicative structure introduced by A. Beurling to study how independent prime number theory is from the additivity of the natural numbers. The results and techniques of this theory apply to other systems having the character of prime numbers and integers; for example, it is used in the study of the prime number theorem (PNT) for ideals of algebraic number fields.
Using both analytic and elementary methods, this book presents many old and new theorems, including several of the authors' results, and many examples of extremal behavior of gnumber systems. Also, the authors give detailed accounts of the \(L^2\) PNT theorem of J. P. Kahane and of the example created with H. L. Montgomery, showing that additive structure is needed for proving the Riemann hypothesis.
Other interesting topics discussed are propositions “equivalent” to the PNT, the role of multiplicative convolution and Chebyshev's prime number formula for gnumbers, and how Beurling theory provides an interpretation of the smooth number formulas of Dickman and de Bruijn.
Graduate students and researchers interested in analytic number theory.

Chapters

Chapter 1. Overview

Chapter 2. Analytic machinery

Chapter 3. $\mathbf {dN}$ as an exponential and Chebyshev’s identity

Chapter 4. Upper and lower estimates of $\mathbf {N(x)}$

Chapter 5. Mertens’ formulas and logarithmic density

Chapter 6. Odensity of gintegers

Chapter 7. Density of gintegers

Chapter 8. Simple estimates of $\mathbf {\pi (x)}$

Chapter 9. Chebyshev bounds—Elementary theory

Chapter 10. WienerIkehara Tauberian theorems

Chapter 11. Chebyshev bounds—Analytic methods

Chapter 12. Optimality of a Chebyshev bound

Chapter 13. Beurling’s PNT

Chapter 14. Equivalences to the PNT

Chapter 15. Kahane’s PNT

Chapter 16. PNT with remainder

Chapter 17. Optimality of the dlVP remainder term

Chapter 18. The Dickman and Buchstab functions

The book is very well written and should prove accessible to a fairly large audience, requiring only some familiarity with mathematical analysis and classical techniques from analytic number theory.
JeanMarie De Koninck, Mathematical Reviews