Beurling Generalized Numbers
Seiten
2016
American Mathematical Society (Verlag)
978-1-4704-3045-0 (ISBN)
American Mathematical Society (Verlag)
978-1-4704-3045-0 (ISBN)
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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 behaviour of g-number systems. The authors also 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.
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.
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.
Harold G. Diamond, University of Illinois, Urbana, IL, USA. Wen-Bin Zhang (Cheung Man Ping), University of the West Indies, Kingston, Jamaica.
Overview
Analytic machinery
$/mathbf{dN}$ as an exponential and Chebyshev's identity
Upper and lower estimates of $/mathbf{N(x)}$
Mertens' formulas and logarithmic density
O-density of g-integers
Density of g-integers
Simple estimates of $/mathbf{/pi(x)}$
Chebyshev bounds--Elementary theory
Wiener-Ikehara Tauberian theorems
Chebyshev bounds--Analytic methods
Optimality of a Chebyshev bound
Beurling's PNT
Equivalences to the PNT
Kahane's PNT
PNT with remainder
Optimality of the dlVP remainder term
The Dickman and Buchstab functions
Bibliography
Index
| Erscheinungsdatum | 10.11.2016 |
|---|---|
| Reihe/Serie | Mathematical Surveys and Monographs |
| Mitarbeit |
Sonstige Mitarbeit: Wen-Bin Zhang |
| Verlagsort | Providence |
| Sprache | englisch |
| Maße | 178 x 254 mm |
| Gewicht | 615 g |
| Themenwelt | Mathematik / Informatik ► Mathematik ► Algebra |
| Mathematik / Informatik ► Mathematik ► Arithmetik / Zahlentheorie | |
| ISBN-10 | 1-4704-3045-2 / 1470430452 |
| ISBN-13 | 978-1-4704-3045-0 / 9781470430450 |
| Zustand | Neuware |
| Informationen gemäß Produktsicherheitsverordnung (GPSR) | |
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