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Beurling Generalized Numbers [Hardback]

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  • Formāts: Hardback, 244 pages, height x width: 254x178 mm, weight: 615 g
  • Sērija : Mathematical Surveys and Monographs
  • Izdošanas datums: 01-Sep-2016
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470430452
  • ISBN-13: 9781470430450
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Beurling Generalized NumbersPalielināt
  • Formāts: Hardback, 244 pages, height x width: 254x178 mm, weight: 615 g
  • Sērija : Mathematical Surveys and Monographs
  • Izdošanas datums: 01-Sep-2016
  • Izdevniecība: American Mathematical Society
  • ISBN-10: 1470430452
  • ISBN-13: 9781470430450
Citas grāmatas par šo tēmu:
Permanent link: https://www.kriso.lv/db/9781470430450.html
Keywords:
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.
Preface xi
Chapter 1 Overview
1(8)
1.1 Some questions about primes
1(1)
1.2 The cast
2(2)
1.3 Examples
4(2)
1.4 π, II, and an extended notion of g-numbers
6(1)
1.5 Notes
7(2)
Chapter 2 Analytic Machinery
9(10)
2.1 A function class
9(1)
2.2 Measures
9(1)
2.3 Mellin transforms
10(1)
2.4 Norms
11(1)
2.5 Convergence
11(1)
2.6 Convolution of measures
12(3)
2.7 Convolution of functions
15(1)
2.8 The L and T operators
16(2)
2.9 Notes
18(1)
Chapter 3 dN as an Exponential and Chebyshev's Identity
19(10)
3.1 Goals and plan
19(2)
3.2 Power series in measures
21(1)
3.3 Inverses
22(1)
3.4 The exponential on V
23(4)
3.5 Three equivalent formulas
27(1)
3.6 Notes
28(1)
Chapter 4 Upper and Lower Estimates of N(x)
29(12)
4.1 Normalization and restriction
29(1)
4.2 O-log density
30(3)
4.3 Lower log density
33(3)
4.4 An example with infinite residue but 0 lower log density
36(1)
4.5 Extreme thinness is inherited
37(2)
4.6 Regular growth
39(1)
4.7 Notes
40(1)
Chapter 5 Mertens' Formulas and Logarithmic Density
41(12)
5.1 Introduction
41(1)
5.2 Logarithmic density
41(2)
5.3 The Hardy-Littlewood-Karamata Theorem
43(2)
5.4 Mertens' sum formula
45(1)
5.5 Mertens' product formula
46(1)
5.6 A remark on γ
47(1)
5.7 An equivalent form and proof of "only if"
48(1)
5.8 Tauber's Theorem and conclusion of the argument
49(2)
5.9 Notes
51(2)
Chapter 6 O-Density of g-integers
53(10)
6.1 Non-relation of log-density and O-density
53(3)
6.2 O-Criteria for O-density
56(4)
6.3 Sharper criteria for O-density
60(2)
6.4 Notes
62(1)
Chapter 7 Density of g-integers
63(14)
7.1 Densities and right hand residues
63(1)
7.2 Axer's Theorem
63(2)
7.3 Criteria for density
65(4)
7.4 An L1 criterion for density
69(3)
7.5 Estimates of N(x) with an error term
72(4)
7.6 Notes
76(1)
Chapter 8 Simple Estimates of π(x)
77(6)
8.1 Unboundedness of π(x)
77(1)
8.2 Can there be as many primes as integers?
78(1)
8.3 π(x) estimates via regular growth
79(2)
8.4 Lower bounds for Σ 1/Pi via lower log-density
81(1)
8.5 Notes
81(2)
Chapter 9 Chebyshev Bounds -- Elementary Theory
83(16)
9.1 Introduction
83(1)
9.2 Chebyshev bounds for natural primes
83(4)
9.3 An auxiliary function
87(3)
9.4 Chebyshev bounds for g-primes
90(5)
9.5 A failure of Chebyshev bounds
95(3)
9.6 Notes
98(1)
Chapter 10 Wiener-Ikehara Tauberian Theorems
99(12)
10.1 Introduction
99(1)
10.2 Wiener-Ikehara Theorems
99(2)
10.3 The Fejer kernel
101(3)
10.4 Proof of the Wiener-Ikehara Theorems
104(4)
10.5 A W-I oscillatory example
108(2)
10.6 Notes
110(1)
Chapter 11 Chebyshev Bounds -- Analytic Methods
111(8)
11.1 Introduction
111(1)
11.2 Wiener-Ikehara setup
112(1)
11.3 A first decomposition
113(1)
11.4 Further decomposition of I2, σ(y)
114(2)
11.5 Chebyshev bounds
116(1)
11.6 Notes
117(2)
Chapter 12 Optimality of a Chebyshev Bound
119(14)
12.1 Introduction
119(1)
12.2 The g-prime system PB
119(3)
12.3 Chebyshev bounds and the zeta function ζB(s)
122(2)
12.4 The counting function NB(x)
124(3)
12.5 Fundamental estimates
127(2)
12.6 Proof of the Optimality Theorem
129(2)
12.7 Notes
131(2)
Chapter 13 Beurling's PNT
133(18)
13.1 Introduction
133(1)
13.2 A lower bound for |ζ(σ + it)|
133(2)
13.3 Nonvanishing of ζ(1 + it)
135(2)
13.4 An L1 condition and conclusion of the proof
137(2)
13.5 Optimality -- a continuous example
139(5)
13.6 Optimality -- a discrete example
144(4)
13.7 Notes
148(3)
Chapter 14 Equivalences to the PNT
151(10)
14.1 Introduction
151(1)
14.2 Implications
151(1)
14.3 Sharp Mertens relation and the PNT
152(2)
14.4 Optimality of the sharp Mertens theorem
154(1)
14.5 Implications between M(x) = o(x) and m(x) = o(1)
155(1)
14.6 Connections of the PNT with M(x) = o(x)
156(2)
14.7 Sharp Mertens relation and m(x) = o(1)
158(2)
14.8 Notes
160(1)
Chapter 15 Kahane's PNT
161(14)
15.1 Introduction
161(1)
15.2 Zeros of the zeta function
162(3)
15.3 A lower bound for |ζ(σ + it)|
165(1)
15.4 A Schwartz function and Poisson summation
166(4)
15.5 Estimating the sum of a series by an improper integral
170(2)
15.6 Conclusion of the proof
172(2)
15.7 Notes
174(1)
Chapter 16 PNT with Remainder
175(20)
16.1 Introduction
175(1)
16.2 Two general lemmas
176(3)
16.3 A Nyman type remainder term
179(7)
16.4 A dlVP-type remainder term
186(6)
16.5 Notes
192(3)
Chapter 17 Optimality of the dlVP Remainder Term
195(34)
17.1 Background
195(1)
17.2 Discrete random approximation
196(8)
17.3 Generalized primes satisfying the Riemann Hypothesis
204(4)
17.4 Generalized primes with large oscillation
208(1)
17.5 Properties of G(z)
209(1)
17.6 Representation of log G(z) as a Mellin transform
210(4)
17.7 A template zeta function
214(4)
17.8 Asymptotics of Nb(x)
218(4)
17.9 Asymptotics of ψB(x)
222(4)
17.10 Normalization and hybrid
226(1)
17.11 Notes
227(2)
Chapter 18 The Dickman and Buchstab Functions
229(10)
18.1 Introduction
229(1)
18.2 The ψ(a, y) function
230(3)
18.3 The φ(x, y) function
233(1)
18.4 A Beurling version of ψ(x, y)
234(1)
18.5 G-numbers with primes from an interval
235(2)
18.6 Other relations
237(1)
18.7 Notes
238(1)
Bibliography 239(4)
Index 243
Harold G. Diamond, University of Illinois, Urbana, IL, USA.

Wen-Bin Zhang (Cheung Man Ping), University of the West Indies, Kingston, Jamaica.