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Beurling Generalized Numbers

Harold G. Diamond University of Illinois, Urbana, IL
Wen-Bin Zhang (Cheung Man Ping) University of the West Indies, Kingston, Jamaica
Available Formats:
Hardcover ISBN: 978-1-4704-3045-0
Product Code: SURV/213
List Price: $110.00 MAA Member Price:$99.00
AMS Member Price: $88.00 Electronic ISBN: 978-1-4704-3538-7 Product Code: SURV/213.E List Price:$110.00
MAA Member Price: $99.00 AMS Member Price:$88.00
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List Price: $165.00 MAA Member Price:$148.50
AMS Member Price: $132.00 Click above image for expanded view Beurling Generalized Numbers Harold G. Diamond University of Illinois, Urbana, IL Wen-Bin Zhang (Cheung Man Ping) University of the West Indies, Kingston, Jamaica Available Formats:  Hardcover ISBN: 978-1-4704-3045-0 Product Code: SURV/213  List Price:$110.00 MAA Member Price: $99.00 AMS Member Price:$88.00
 Electronic ISBN: 978-1-4704-3538-7 Product Code: SURV/213.E
 List Price: $110.00 MAA Member Price:$99.00 AMS Member Price: $88.00 Bundle Print and Electronic Formats and Save! This product is available for purchase as a bundle. Purchasing as a bundle enables you to save on the electronic version.  List Price:$165.00 MAA Member Price: $148.50 AMS Member Price:$132.00
• Book Details

Mathematical Surveys and Monographs
Volume: 2132016; 244 pp
MSC: 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 g-number 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 g-numbers, 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. O-density of g-integers
• Chapter 7. Density of g-integers
• Chapter 8. Simple estimates of $\mathbf {\pi (x)}$
• Chapter 9. Chebyshev bounds—Elementary theory
• Chapter 10. Wiener-Ikehara 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

• 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.

Jean-Marie De Koninck, Mathematical Reviews
• Requests

Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
Volume: 2132016; 244 pp
MSC: 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 g-number 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 g-numbers, 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. O-density of g-integers
• Chapter 7. Density of g-integers
• Chapter 8. Simple estimates of $\mathbf {\pi (x)}$
• Chapter 9. Chebyshev bounds—Elementary theory
• Chapter 10. Wiener-Ikehara 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.

Jean-Marie De Koninck, Mathematical Reviews
Review Copy – for reviewers who would like to review an AMS book
Permission – for use of book, eBook, or Journal content
Accessibility – to request an alternate format of an AMS title
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