Beurling generalized numbers / Harold G. Diamond, Wen-Bin Zhang (Cheung Man Ping).
Material type:
TextSeries: Mathematical surveys and monographs ; v 213.Publication details: Providence : American Mathematical Society, ©2016.Description: xi, 244 pages ; 26 cmISBN: - 9781470430450
- 510MS 23 Am512
| Item type | Current library | Call number | Status | Date due | Barcode | Item holds | |
|---|---|---|---|---|---|---|---|
| Books | ISI Library, Kolkata | 510MS Am512 (Browse shelf(Opens below)) | Available | 137676 |
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| 510MS Am512 Dynamical Mordell-Lang conjecture / | 510MS Am512 Galois theories of linear difference equations : an introduction / | 510MS Am512 Ramsey theory for product spaces / | 510MS Am512 Beurling generalized numbers / | 510MS Am512 Shock formation in small-data solutions to 3D quasilinear wave equations / | 510MS Am512 On groups of PL-homeomorphisms of the real line / | 510MS Am512 Study in derived algebraic geometry : |
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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