Quantized Number Theory, Fractal Strings and the Riemann Hypothesis: From Spectral Operators to Phase Transitions and Universality

Contents
Overview
Preface
List of Figures
List of Tables
Conventions
1 Introduction
2 Generalized Fractal Strings and Complex Dimensions
2.1 Ordinary and generalized fractal strings
2.2 Harmonic string and spectral measure
2.3 Explicit formulas
3 Direct and Inverse Spectral Problems for Fractal Strings
3.1 Minkowski dimension and Minkowski measurability criteria
3.2 Direct spectral problems and the (modified) Weyl–Berry conjecture
3.3 Inverse spectral problems and the Riemann hypothesis
4 The Heuristic Spectral Operator ac
4.1 The heuristic multiplicative and additive spectral operators
4.2 The heuristic spectral operator and its Euler product
5 The Infinitesimal Shift ∂c
5.1 The weighted Hilbert space Hc
5.2 The domain of the differentiation operator ∂c
5.3 Normality of the infinitesimal shift ∂c
6 The Spectrum of the Infinitesimal Shift ∂c
6.1 Characterization of the spectrum of an unbounded normal operator
6.2 The spectrum of the differentiation operator ∂c
6.3 The shift group {e−t∂c}t∈R and the infinitesimal shift ∂c
6.4 The truncated infinitesimal shifts ∂(T)c,± and their spectra
6.4.1 The continuous case
6.4.2 The meromorphic case
7 The Spectral Operator ac = ζ(∂c): Quantized Dirichlet Series, Euler Product, and Analytic Continuation
7.1 Overview
7.2 Rigorous definition of the spectral operator: ac = ζ(∂c)
7.3 Quantized Dirichlet series (case c > 1)
7.4 Quantized Euler product (case c > 1)
7.5 Further justification of the definition of ac: Operator-valued analytic continuation (case c > 0)
7.5.1 Analytic continuation of the spectral operator
7.5.2 Convergent quantized Dirichlet series and Euler product in the critical interval 0 < c < 1
7.5.3 Conjecture and open problem about ac (for 0 < c < 1)
7.6 The global spectral operator Ac = ξ(∂c): Quantized analytic continuation (for c ∈ R) and functional equation
7.6.1 The global Riemann zeta function ξ = ξ(s) (for s ∈ C)
7.6.2 Definition and operator-valued analytic continuation of Ac = ξ(∂c) (for c ∈ R)
7.6.3 Quantized functional equation
8 Spectral Reformulation of the Riemann Hypothesis
8.1 The truncated spectral operators a(T)c,± = ζ(∂(T)c,±) and their spectra
8.2 Quasi-invertibility of ac and the Riemann zeros
8.3 Inverse spectral problems for fractal strings and a spectral reformulation of the Riemann hypothesis
8.4 Almost invertibility of ac and an almost Riemann hypothesis
9 Zeta Values, Riemann Zeros and Phase Transitions for ac = ζ(∂c)
9.1 The spectrum of ac
9.2 Invertibility of ac and zeta values
9.3 Phase transitions of ac at c = 1/2 and c = 1
9.3.1 Phase transitions for the boundedness and invertibility of ac
9.3.2 Phase transitions for the shape of the spectrum of ac
9.3.3 Possible interpretations of the phase transitions
9.4 An asymmetric criterion for the Riemann hypothesis: Invertibility of ac for 0 < c < 1/2
10 A Quantum Analog of the Universality of ζ(s)
10.1 Universality of the Riemann zeta function ζ = ζ(s)
10.1.1 Origins of universality
10.1.2 Voronin’s universality theorem
10.2 Universality and an operator-valued extended Voronin theorem
10.2.1 A first quantum analog of the universality theorem
10.2.2 A more general version of quantized universality
11 Concluding Comments and Future Research Directions
A Riemann’s Explicit Formula
B Natural Boundary Conditions for ∂c
C The Momentum Operator and Normality of ∂c
D The Spectral Mapping Theorem
E The Range and Growth of ζ(s) on Vertical Lines
E.1 Estimates for the modulus of ζ(s) along vertical lines within the critical strip
E.2 The range of ζ(s) and its Euler factors to the right of the critical strip
F Further Extensions of the Universality of ζ(s)
F.1 A first extension of Voronin’s original theorem
F.2 Extensions to certain L-functions and other zeta functions
Acknowledgments
Bibliography
Index of Symbols
Author Index
Subject Index