Beurling Generalized Numbers (Mathematical Surveys and Monographs) (Mathematical Surveys and Monographs, 213)

✍ Scribed by Harold G. Diamond, Wen-Bin Zhang

• Publisher: American Mathematical Society
• Year: 2016
• Tongue: English
• Leaves: 258
• Category: Library

✦ Synopsis

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.

✦ Table of Contents

Cover
Title page
Contents
Preface
Chapter 1. Overview
1.1. Some questions about primes
1.2. The cast
1.3. Examples
1.4. 𝜋,Π, and an extended notion of g-numbers
1.5. Notes
Chapter 2. Analytic Machinery
2.1. A function class
2.2. Measures
2.3. Mellin transforms
2.4. Norms
2.5. Convergence
2.6. Convolution of measures
2.7. Convolution of functions
2.8. The 𝐋 and 𝐓 operators
2.9. Notes
Chapter 3. 𝑑𝑁 as an Exponential and Chebyshev’s Identity
3.1. Goals and plan
3.2. Power series in measures
3.3. Inverses
3.4. The exponential on \cV
3.5. Three equivalent formulas
3.6. Notes
Chapter 4. Upper and Lower Estimates of 𝑁(𝑥)
4.1. Normalization and restriction
4.2. O-log density
4.3. Lower log density
4.4. An example with infinite residue but 0 lower log density
4.5. Extreme thinness is inherited
4.6. Regular growth
4.7. Notes
Chapter 5. Mertens’ Formulas and Logarithmic Density
5.1. Introduction
5.2. Logarithmic density
5.3. The Hardy-Littlewood-Karamata Theorem
5.4. Mertens’ sum formula
5.5. Mertens’ product formula
5.6. A remark on 𝛾
5.7. An equivalent form and proof of “only if”
5.8. Tauber’s Theorem and conclusion of the argument
5.9. Notes
Chapter 6. O-Density of g-integers
6.1. Non-relation of log-density and O-density
6.2. O-Criteria for O-density
6.3. Sharper criteria for O-density
6.4. Notes
Chapter 7. Density of g-integers
7.1. Densities and right hand residues
7.2. Axer’s Theorem
7.3. Criteria for density
7.4. An 𝐿¹ criterion for density
7.5. Estimates of 𝑁(𝑥) with an error term
7.6. Notes
Chapter 8. Simple Estimates of 𝜋(𝑥)
8.1. Unboundedness of 𝜋(𝑥)
8.2. Can there be as many primes as integers?
8.3. 𝜋(𝑥) estimates via regular growth
8.4. Lower bounds for ∑1/𝑝ᵢ via lower log-density
8.5. Notes
Chapter 9. Chebyshev Bounds –Elementary Theory
9.1. Introduction
9.2. Chebyshev bounds for natural primes
9.3. An auxiliary function
9.4. Chebyshev bounds for g-primes
9.5. A failure of Chebyshev bounds
9.6. Notes
Chapter 10. Wiener-Ikehara Tauberian Theorems
10.1. Introduction
10.2. Wiener-Ikehara Theorems
10.3. The Fejér kernel
10.4. Proof of the Wiener-Ikehara Theorems
10.5. A W-I oscillatory example
10.6. Notes
Chapter 11. Chebyshev Bounds –Analytic Methods
11.1. Introduction
11.2. Wiener-Ikehara setup
11.3. A first decomposition
11.4. Further decomposition of 𝐼_{2,𝜎}(𝑦)
11.5. Chebyshev bounds
11.6. Notes
Chapter 12. Optimality of a Chebyshev Bound
12.1. Introduction
12.2. The g-prime system 𝒫_{ℬ}
12.3. Chebyshev bounds and the zeta function 𝜁_{𝐵}(𝑠)
12.4. The counting function 𝑁_{𝐵}(𝑥)
12.5. Fundamental estimates
12.6. Proof of the Optimality Theorem
12.7. Notes
Chapter 13. Beurling’s PNT
13.1. Introduction
13.2. A lower bound for |𝜁(\s+𝑖𝑡)|
13.3. Nonvanishing of 𝜁(1+𝑖𝑡)
13.4. An 𝐿¹ condition and conclusion of the proof
13.5. Optimality –a continuous example
13.6. Optimality –a discrete example
13.7. Notes
Chapter 14. Equivalences to the PNT
14.1. Introduction
14.2. Implications
14.3. Sharp Mertens relation and the PNT
14.4. Optimality of the sharp Mertens theorem
14.5. Implications between 𝑀(𝑥)=𝑜(𝑥) and 𝑚(𝑥)=𝑜(1)
14.6. Connections of the PNT with 𝑀(𝑥)=𝑜(𝑥)
14.7. Sharp Mertens relation and 𝑚(𝑥)=𝑜(1)
14.8. Notes
Chapter 15. Kahane’s PNT
15.1. Introduction
15.2. Zeros of the zeta function
15.3. A lower bound for |𝜁(𝜎+𝑖𝑡)|
15.4. A Schwartz function and Poisson summation
15.5. Estimating the sum of a series by an improper integral
15.6. Conclusion of the proof
15.7. Notes
Chapter 16. PNT with Remainder
16.1. Introduction
16.2. Two general lemmas
16.3. A Nyman type remainder term
16.4. A dlVP-type remainder term
16.5. Notes
Chapter 17. Optimality of the dlVP Remainder Term
17.1. Background
17.2. Discrete random approximation
17.3. Generalized primes satisfying the Riemann Hypothesis
17.4. Generalized primes with large oscillation
17.5. Properties of 𝐺(𝑧)
17.6. Representation of log𝐺(𝑧) as a Mellin transform
17.7. A template zeta function
17.8. Asymptotics of 𝑁_{𝐵}(𝑥)
17.9. Asymptotics of 𝜓_{𝐵}(𝑥)
17.10. Normalization and hybrid
17.11. Notes
Chapter 18. The Dickman and Buchstab Functions
18.1. Introduction
18.2. The 𝜓(𝑥,𝑦) function
18.3. The 𝜑(𝑥,𝑦) function
18.4. A Beurling version of 𝜓(𝑥,𝑦)
18.5. G-numbers with primes from an interval
18.6. Other relations
18.7. Notes
Bibliography
Index
Back Cover