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Beurling generalized numbers / Harold G. Diamond, Wen-Bin Zhang (Cheung Man Ping).

By: Contributor(s): Material type: TextTextSeries: Mathematical surveys and monographs ; v 213.Publication details: Providence : American Mathematical Society, ©2016.Description: xi, 244 pages ; 26 cmISBN:
  • 9781470430450
Subject(s): DDC classification:
  • 510MS 23 Am512
Contents:
1. Overview -- 2. Analytic machinery -- 3. dN as an exponential and Chebyshevs's identity -- 4. Upper and lower estimates of N (x) -- 5. Mertens' formulas and logarithmic density -- 6. O-Density of g-integers -- 7. Density of g-integers -- 8. Simple estimates of pie (x) -- 9. Chebyshev bounds-elementary theory -- 10. Wiener-Ikehara tauberian theorems -- 11. Chebyshev bounds-analytic methods -- 12. Optimality of a Chebyshev bound -- 13. Beurling's PNT -- 14. Equivalences to th PNT -- 15. Kahane's PNT -- 16. PNT with remainder -- 17. Optimality of the dIVP remainder term -- 18. The Dickman and Buchstab functions.
Summary: 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.
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Holdings
Item type Current library Call number Status Date due Barcode Item holds
Books ISI Library, Kolkata 510MS Am512 (Browse shelf(Opens below)) Available 137676
Total holds: 0

Includes bibliographical references and index.

1. Overview --
2. Analytic machinery --
3. dN as an exponential and Chebyshevs's identity --
4. Upper and lower estimates of N (x) --
5. Mertens' formulas and logarithmic density --
6. O-Density of g-integers --
7. Density of g-integers --
8. Simple estimates of pie (x) --
9. Chebyshev bounds-elementary theory --
10. Wiener-Ikehara tauberian theorems --
11. Chebyshev bounds-analytic methods --
12. Optimality of a Chebyshev bound --
13. Beurling's PNT --
14. Equivalences to th PNT --
15. Kahane's PNT --
16. PNT with remainder --
17. Optimality of the dIVP remainder term --
18. The Dickman and Buchstab functions.

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.

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