Quantized Number Theory, Fractal Strings and the Riemann Hypothesis: From Spectral Operators to Phase Transitions and Universality (Fractals and Dynamics in Mathematics, Science, and the Arts:)

Studying The Relationship Between The Geometry, Arithmetic And Spectra Of Fractals Has Been A Subject Of Significant Interest In Contemporary Mathematics. This Book Contributes To The Literature On The Subject In Several Different And New Ways. In Particular, The Authors Provide A Rigorous And Detailed Study Of The Spectral Operator, A Map That Sends The Geometry Of Fractal Strings Onto Their Spectrum. To That Effect, They Use And Develop Methods From Fractal Geometry, Functional Analysis, Complex Analysis, Operator Theory, Partial Differential Equations, Analytic Number Theory And Mathematical Physics.originally, M L Lapidus And M Van Frankenhuijsen ''heuristically'' Introduced The Spectral Operator In Their Development Of The Theory Of Fractal Strings And Their Complex Dimensions, Specifically In Their Reinterpretation Of The Earlier Work Of M L Lapidus And H Maier On Inverse Spectral Problems For Fractal Strings And The Riemann Hypothesis.one Of The Main Themes Of The Book Is To Provide A Rigorous Framework Within Which The Corresponding Question ''can One Hear The Shape Of A Fractal String?'' Or, Equivalently, ''can One Obtain Information About The Geometry Of A Fractal String, Given Its Spectrum?'' Can Be Further Reformulated In Terms Of The Invertibility Or The Quasi-invertibility Of The Spectral Operator.the Infinitesimal Shift Of The Real Line Is First Precisely Defined As A Differentiation Operator On A Family Of Suitably Weighted Hilbert Spaces Of Functions On The Real Line And Indexed By A Dimensional Parameter C. Then, The Spectral Operator Is Defined Via The Functional Calculus As A Function Of The Infinitesimal Shift. In This Manner, It Is Viewed As A Natural ''quantum'' Analog Of The Riemann Zeta Function. More Precisely, Within This Framework, The Spectral Operator Is Defined As The Composite Map Of The Riemann Zeta Function With The Infinitesimal Shift, Viewed As An Unbounded Normal Operator Acting On The Above Hilbert Space.it Is Shown That The Quasi-invertibility Of The Spectral Operator Is Intimately Connected To The Existence Of Critical Zeros Of The Riemann Zeta Function, Leading To A New Spectral And Operator-theoretic Reformulation Of The Riemann Hypothesis. Accordingly, The Spectral Operator Is Quasi-invertible For All Values Of The Dimensional Parameter C In The Critical Interval (0,1) (other Than In The Midfractal Case When C =1/2) If And Only If The Riemann Hypothesis (rh) Is True. A Related, But Seemingly Quite Different, Reformulation Of Rh, Due To The Second Author And Referred To As An ''asymmetric Criterion For Rh'', Is Also Discussed In Some Detail: Namely, The Spectral Operator Is Invertible For All Values Of C In The Left-critical Interval (0,1/2) If And Only If Rh Is True.these Spectral Reformulations Of Rh Also Led To The Discovery Of Several ''mathematical Phase Transitions'' In This Context, For The Shape Of The Spectrum, The Invertibility, The Boundedness Or The Unboundedness Of The Spectral Operator, And Occurring Either In The Midfractal Case Or In The Most Fractal Case When The Underlying Fractal Dimension Is Equal To 1/2 Or 1, Respectively. In Particular, The Midfractal Dimension C=1/2 Is Playing The Role Of A Critical Parameter In Quantum Statistical Physics And The Theory Of Phase Transitions And Critical Phenomena.furthermore, The Authors Provide A ''quantum Analog'' Of Voronin''s Classical Theorem About The Universality Of The Riemann Zeta Function. Moreover, They Obtain And Study Quantized Counterparts Of The Dirichlet Series And Of The Euler Product For The Riemann Zeta Function, Which Are Shown To Converge (in A Suitable Sense) Even Inside The Critical Strip.for Pedagogical Reasons, Most Of The Book Is Devoted To The Study Of The Quantized Riemann Zeta Function. However, The Results Obtained In This Monograph Are Expected To Lead To A Quantization Of Most Classic Arithmetic Zeta Functions, Hence, Further ''naturally Quantizing'' Various Aspects Of Analytic Number Theory And Arithmetic Geometry.the Book Should Be Accessible To Experts And Non-experts Alike, Including Mathematics And Physics Graduate Students And Postdoctoral Researchers, Interested In Fractal Geometry, Number Theory, Operator Theory And Functional Analysis, Differential Equations, Complex Analysis, Spectral Theory, As Well As Mathematical And Theoretical Physics. Whenever Necessary, Suitable Background About The Different Subjects Involved Is Provided And The New Work Is Placed In Its Proper Historical Context. Several Appendices Supplementing The Main Text Are Also Included. "Studying the relationship between the geometry, arithmetic and spectra of fractals has been a subject of significant interest in contemporary mathematics. This book contributes to the literature on the subject in several different and new ways. In particular, the authors provide a rigorous and detailed study of the spectral operator, a map that sends the geometry of fractal strings onto their spectrum. To that effect, they use and develop methods from fractal geometry, functional analysis, complex analysis, operator theory, partial differential equations, analytic number theory and mathematical physics. Originally, M. L. Lapidus and M. van Frankenhuijsen 'heuristically' introduced the spectral operator in their development of the theory of fractal strings and their complex dimensions, specifically in their reinterpretation of the earlier work of M. L. Lapidus and H. Maier on inverse spectral problems for fractal strings and the Riemann hypothesis. One of the main themes of the book is to provide a rigorous framework within which the corresponding question "Can one hear the shape of a fractal string?" or, equivalently, "Can one obtain information about the geometry of a fractal string, given its spectrum?" can be further reformulated in terms of the invertibility or the quasi-invertibility of the spectral operator. The infinitesimal shift of the real line is first precisely defined as a differentiation operator on a family of suitably weighted Hilbert spaces of functions on the real line and indexed by a dimensional parameter c. Then, the spectral operator is defined via the functional calculus as a function of the infinitesimal shift. In this manner, it is viewed as a natural 'quantum' analog of the Riemann zeta function. More precisely, within this framework, the spectral operator is defined as the composite map of the Riemann zeta function with the infinitesimal shift, viewed as an unbounded normal operator acting on the above Hilbert space. It is shown that the quasi-invertibilty of the spectral operator is intimately connected to the existence of critical zeros of the Riemann zeta function, leading to a new spectral and operator-theoretic reformulation of the Riemann hypothesis. Accordingly, the spectral operator is quasi-invertible for all values of the dimensional parameter c in the critical interval (0,1) (other than in the midfractal case when c =1/2) if and only if the Riemann hypothesis (RH) is true. A related, but seemingly quite different, reformulation of RH, due to the second author and referred to as an 'asymmetric criterion for RH', is also discussed in some detail: namely, the spectral operator is invertible for all values of c in the left-critical interval (0,1/2) if and only if RH is true. These spectral reformulations of RH also led to the discovery of several 'mathematical phase transitions' in this context, for the shape of the spectrum, the invertibilty, the boundedness or the unboundedness of the spectral operator, and occurring either in the midfractal case or in the most fractal case when the underlying fractal dimension is equal to 1/2 or 1, respectively. In particular, the midfractal dimension c=1/2 is playing the role of a critical parameter in quantum statistical physics and the theory of phase transitions and critical phenomena. Furthermore, the authors provide a 'quantum analog' of Voronin's classical theorem about the universality of the Riemann zeta function. Moreover, they obtain and study quantized counterparts of the Dirichlet series and of the Euler product for the Riemann zeta function, which are shown to converge (in a suitable sense) even inside the critical strip. For pedagogical reasons, most of the book is devoted to the study of the quantized Riemann zeta function. However, the results obtained in this monograph are expected to lead to a quantization of most classic arithmetic zeta functions, hence, further 'naturally quantizing' various aspects of analytic number theory and arithmetic geometry. The book should be accessible to experts and non-experts alike, including mathematics and physics graduate students and postdoctoral researchers, interested in fractal geometry, number theory, operator theory and functional analysis, differential equations, complex analysis, spectral theory, as well as mathematical and theoretical physics. Whenever necessary, suitable background about the different subjects involved is provided and the new work is placed in its proper historical context. Several appendices supplementing the main text are also included"-- Provided by publisher Contents 26 Overview 10 Preface 14 List of Figures 30 List of Tables 34 Conventions 36 1 Introduction 42 2 Generalized Fractal Strings and Complex Dimensions 60 2.1 Ordinary and generalized fractal strings 61 2.2 Harmonic string and spectral measure 66 2.3 Explicit formulas 71 3 Direct and Inverse Spectral Problems for Fractal Strings 78 3.1 Minkowski dimension and Minkowski measurability criteria 80 3.2 Direct spectral problems and the (modified) Weyl–Berry conjecture 85 3.3 Inverse spectral problems and the Riemann hypothesis 90 4 The Heuristic Spectral Operator ac 96 4.1 The heuristic multiplicative and additive spectral operators 97 4.2 The heuristic spectral operator and its Euler product 100 5 The Infinitesimal Shift ∂c 106 5.1 The weighted Hilbert space Hc 107 5.2 The domain of the differentiation operator ∂c 112 5.3 Normality of the infinitesimal shift ∂c 119 6 The Spectrum of the Infinitesimal Shift ∂c 134 6.1 Characterization of the spectrum of an unbounded normal operator 136 6.2 The spectrum of the differentiation operator ∂c 148 6.3 The shift group {e−t∂c}t∈R and the infinitesimal shift ∂c 161 6.4 The truncated infinitesimal shifts ∂(T)c,± and their spectra 167 6.4.1 The continuous case 167 6.4.2 The meromorphic case 180 7 The Spectral Operator ac = ζ(∂c): Quantized Dirichlet Series, Euler Product, and Analytic Continuation 190 7.1 Overview 193 7.2 Rigorous definition of the spectral operator: ac = ζ(∂c) 196 7.3 Quantized Dirichlet series (case c > 1) 199 7.4 Quantized Euler product (case c > 1) 211 7.5 Further justification of the definition of ac: Operator-valued analytic continuation (case c > 0) 221 7.5.1 Analytic continuation of the spectral operator 222 7.5.2 Convergent quantized Dirichlet series and Euler product in the critical interval 0 < c < 1 234 7.5.3 Conjecture and open problem about ac (for 0 < c < 1) 241 7.6 The global spectral operator Ac = ξ(∂c): Quantized analytic continuation (for c ∈ R) and functional equation 244 7.6.1 The global Riemann zeta function ξ = ξ(s) (for s ∈ C) 244 7.6.2 Definition and operator-valued analytic continuation of Ac = ξ(∂c) (for c ∈ R) 247 7.6.3 Quantized functional equation 251 8 Spectral Reformulation of the Riemann Hypothesis 256 8.1 The truncated spectral operators a(T)c,± = ζ(∂(T)c,±) and their spectra 258 8.2 Quasi-invertibility of ac and the Riemann zeros 261 8.3 Inverse spectral problems for fractal strings and a spectral reformulation of the Riemann hypothesis 269 8.4 Almost invertibility of ac and an almost Riemann hypothesis 271 9 Zeta Values, Riemann Zeros and Phase Transitions for ac = ζ(∂c) 276 9.1 The spectrum of ac 278 9.2 Invertibility of ac and zeta values 281 9.3 Phase transitions of ac at c = 1/2 and c = 1 283 9.3.1 Phase transitions for the boundedness and invertibility of ac 284 9.3.2 Phase transitions for the shape of the spectrum of ac 286 9.3.3 Possible interpretations of the phase transitions 292 9.4 An asymmetric criterion for the Riemann hypothesis: Invertibility of ac for 0 < c < 1/2 294 10 A Quantum Analog of the Universality of ζ(s) 320 10.1 Universality of the Riemann zeta function ζ = ζ(s) 321 10.1.1 Origins of universality 321 10.1.2 Voronin’s universality theorem 323 10.2 Universality and an operator-valued extended Voronin theorem 328 10.2.1 A first quantum analog of the universality theorem 328 10.2.2 A more general version of quantized universality 333 11 Concluding Comments and Future Research Directions 338 A Riemann’s Explicit Formula 346 B Natural Boundary Conditions for ∂c 352 C The Momentum Operator and Normality of ∂c 356 D The Spectral Mapping Theorem 364 E The Range and Growth of ζ(s) on Vertical Lines 376 E.1 Estimates for the modulus of ζ(s) along vertical lines within the critical strip 378 E.2 The range of ζ(s) and its Euler factors to the right of the critical strip 385 F Further Extensions of the Universality of ζ(s) 390 F.1 A first extension of Voronin’s original theorem 390 F.2 Extensions to certain L-functions and other zeta functions 392 Acknowledgments 396 Bibliography 398 Index of Symbols 428 Author Index 440 Subject Index 450