Quantized Number Theory, Fractal Stringsand the Riemann Hypothesis

From Spectral Operators

to Phase Transitions and Universality

Hafedh Herichi and Michel L. Lapidus

2

0.1 Overview

The spectral operator was introduced heuristically by M.L. Lapidusand M. van Frankenhuijsen in their reinterpretation of the earlier work ofM. L. Lapidus and H.Maier [LapMa2] on inverse spectral problems for frac-tal strings and the Riemann hypothesis. In essence, it is a map that sendsthe geometry of a fractal string onto its spectrum. In this book,we providea rigorous functional analytic framework within which to study the spectraloperator. Namely, we introduce an appropriate weighted Hilbert space Hc,depending on a parameter c ∈ R.When c lies in the critical interval (0, 1),it can be thought of as being directly related to the fractal (i.e., Minkowskior box) dimension of the underlying fractal strings.

The spectral operator ac itself can be viewed as a suitable ‘quantization’of the celebrated Riemann zeta function ζ(s). In fact, it is precisely definedhere as the composite map of ζ(s) and of the infinitesimal shift (of the realline) ∂c =

ddt : ac = ζ(∂c), acting on the Hilbert space Hc = L2(R, e−2ctdt).

We show that the infinitesimal shift ∂c is an unbounded normal opera-tor on Hc, with spectrum the vertical line Re(s) = c. Applying the functionalcalculus for unbounded normal operators combined with a suitable versionof the spectral mapping theorem, we deduce that ac is well defined and thatits spectrum is equal to the closure of the range of ζ(s) on the vertical lineRe(s) = c. Furthermore, we show that ∂c is indeed the infinitesimal shiftof the real line acting on Hc, and we study the associated (shift) group ofoperators on Hc, for any c ∈ R.Moreover, (If we truncate ∂c appropriately,and avoid the value c = 1 corresponding to the pole of ζ(s) at s = 1, then

the associated ‘truncated spectral operator’ a(T )c = ζ(∂

(T )c ) is shown to have

for spectrum the range of ζ(s) on [c− iT, c+ iT ], for each T > 0.

For every c > 1, we show that the spectral operator ac coincideswith the corresponding operator-valued Dirichlet series

∑∞n=1 n

−∂c and Eu-ler product

∏p∈P(1 − p−∂c)−1, where P denotes the set of all primes.We

also obtain quantum analogs of the classic analytic continuation of ζ(s) inthe half-plane Re(s) > 0 and in the whole complex plane C (i.e., we ob-tain ‘analytic continuation’ representations of ac for c > 0 and for c ∈ R,respectively). In the process, we obtain and study a natural quantum (i.e.,operator-valued) analog Ac = ξ(∂) of the completed (or global) Riemannzeta function ξ(s) = π−

s2 γ( s2)ζ(s) and the associated functional equation.

i

We then apply this theory to obtain a necessary and sufficient condi-tion for the invertibility (in a suitable sense) of the spectral operator (in thecritical strip) and therefore obtain a new spectral and operator-theoretic re-formulation of the Riemann hypothesis.More specifically, we show that thespectral operator is quasi-invertible (i.e., the truncated spectral operator

a(T )c = ζ(∂

(T )c ) is invertible, for every T > 0) if and only if ζ(s) does not

have any zeros on the vertical line Re(s) = c. Hence, it is not quasi-invertiblein the midfractal case where c = 1

2 , and it is quasi-invertible everywhere else(i.e., for all c ∈ (0, 1) with c 6= 1

2 ) if and only if the Riemann hypothesis istrue.This result sheds new light on the earlier reformulation of the Riemannhypothesis (expressed in terms of an inverse spectral problem for vibratingfractal strings) obtained by M. L. Lapidus and H.Maier in [LapMa1–2] andlater revisited (in terms of the notion of complex fractal dimensions and ageneralization of Riemann’s explicit formula) by M.L. Lapidus and M.vanFrankenhuijsen in [Lap-vF2–4].

Moreover, we show the existence of several mathematical phase tran-sitions for the spectral operator at the critical fractal dimensions c = 1

2 andc = 1, concerning the shape of the spectrum of ac, its boundedness, its invert-ibility as well as its quasi-invertibility.We also discuss possible mathemati-cal and physical interpretations of the phase transitions of ac. Furthermore,we provide an ‘asymmetric criterion’ for the Riemann hypothesis (RH) bystudying the invertibility of ac for 0 < c < 1

2 .More specifically, it is shownthat the spectral operator ac is invertible (in the usual sense of unboundedoperators) for all c ∈ (0, 12) if and only if RH is true. (This is not equiva-lent to the analogous statement for all c ∈ (12 , 1), which is not true for anyvalue of the parameter c in (0, 12).) This asymmetric reformulation of RHwas obtained earlier by the second author, using some of the results of thepresent work. It relies on the non-universality of ζ(s) in the left critical strip0 < Re(s) < 1

2 (i.e., for 0 < c < 12 ) and its asymmetry comes from the uni-

versality of ζ(s) in the right critical strip 12 < Re(s) < 1 (i.e., for 1

2 < c < 1).

Moreover, by rigorously studying the family of truncated infinitesimalshifts and their corresponding truncated operator-valued spectral operators,we provide a ‘quantum analog’ of the universality of the Riemann zeta func-tion obtained by S.M.Voronin. It follows that the spectral operator is chaoticand hence, that it can emulate any type of complex behavior. (Roughlyspeaking, the classic universality theorem states that any nowwhere vanish-ing holomorphic function in a suitable bounded open subset of the critical

ii

strip 12 < Re(s) < 1 can be uniformly approximated by suitable vertical

translates of ζ(s).)

Finally, we discuss various open problems and propose several direc-tions for future research concerning, in particular, the extension of the the-ory to the quantization of general L-functions, the associated Euler productand Dirichlet series representations, as well as the corresponding global (orcompleted) arithmetic zeta functions. These proposed extensions, along withthe results established in this book, should play a key role in further ‘natu-rally’ quantizing various aspects of analytic number theory and arithmeticgeometry.

iii

0.2 Preface

iv

0.3 List of Figures

v

0.4 List of Tables

vi

Contents

0.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . i

0.2 Preface . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . iv

0.3 List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . v

0.4 List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . vi

1 Introduction 1

2 Generalized Fractal Strings and Complex Dimensions 19

2.1 Ordinary and generalized fractal strings . . . . . . . . . . . . 20

2.2 Harmonic string and spectral measure . . . . . . . . . . . . . 23

2.3 Explicit formulas . . . . . . . . . . . . . . . . . . . . . . . . . 26

3 Direct and Inverse Spectral Problems for Fractal Strings 31

3.1 Minkowski dimension and Minkowski measurability criteria . 33

3.2 Direct spectral problems and the (modified) Weyl–Berry con-jecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 37

3.3 Inverse spectral problems and the Riemann hypothesis . . . . 42

4 The Heuristic Spectral Operator ac 47

4.1 The heuristic multiplicative and additive spectral operators . 48

4.2 The heuristic spectral operator and its Euler product . . . . . 50

5 The Infinitesimal Shift ∂c 55

5.1 The weighted Hilbert space Hc . . . . . . . . . . . . . . . . . 56

5.2 The domain of the differentiation operator ∂c . . . . . . . . . 61

5.3 Normality of the infinitesimal shift ∂c . . . . . . . . . . . . . 67

6 The Spectrum of the Infinitesimal Shift ∂c 81

6.1 Characterization of the spectrum of an unbounded normaloperator . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 83

6.2 The spectrum of the differentiation operator ∂c . . . . . . . . 87

vii

6.3 The shift group {e−t∂c}t∈R and the infinitesimal shift ∂c . . . 93

6.4 The truncated infinitesimal shifts ∂(T )c and their spectra . . . 98

6.5 The continuous case . . . . . . . . . . . . . . . . . . . . . . . 98

6.6 The meromorphic case . . . . . . . . . . . . . . . . . . . . . . 104

7 Precise Definition of the Spectral Operator: ac = ζ(∂c). QuantizedDirichlet Series, Euler Product, and Analytic Continuation107

7.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107

7.2 Rigorous definition of ac . . . . . . . . . . . . . . . . . . . . . 111

7.3 Quantized Dirichlet series (case c > 1) . . . . . . . . . . . . . 114

7.4 Quantized Euler product (case c > 1) . . . . . . . . . . . . . . 126

7.5 Further justification of the definition of ac: Operator-valuedanalytic continuation (case c > 0) . . . . . . . . . . . . . . . . 134

7.6 Analytic continuation of the spectral operator . . . . . . . . . 135

7.7 Global spectral operator: Quantized analytic continuation(for c ∈ R) and functional equation . . . . . . . . . . . . . . . 153

7.8 Truncated spectral operators: Quantized Dirichlet series, Eu-ler product and analytic continuation (cases c > 0 and c ∈ R) 153

8 A Spectral Reformulation of the Riemann Hypothesis 155

8.1 The truncated spectral operators a(T )c = ζ(∂

(T )c ) and their

spectra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 156

8.2 Quasi-invertibility of ac and the Riemann zeros . . . . . . . . 159

8.3 Inverse spectral problems for fractal strings and a reformula-tion of the Riemann hypothesis . . . . . . . . . . . . . . . . . 163

8.4 Almost invertibility of ac and an almost Riemann hypothesis 164

9 Zeta Values, Riemann Zeros and Phase Transitions for ac =ζ(∂c) 169

9.1 The spectrum of ac . . . . . . . . . . . . . . . . . . . . . . . . 171

9.2 Invertibility of ac and zeta values . . . . . . . . . . . . . . . . 173

9.3 Phase transitions of ac at c =12 and c = 1 . . . . . . . . . . . 174

9.3.1 Phase transitions for the boundedness and invertibil-ity of ac . . . . . . . . . . . . . . . . . . . . . . . . . . 174

9.3.2 Phase transitions for the shape of the spectrum of ac . 176

9.3.3 Possible interpretations of the phase transitions . . . . 179

9.4 An asymmetric criterion for the Riemann hypothesis: Invert-ibility of ac for 0 < c < 1

2 . . . . . . . . . . . . . . . . . . . . 180

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10 A Quantum Analog of the Universality of ζ(s) 18310.1 Universality of the Riemann zeta function ζ = ζ(s) . . . . . . 183

10.1.1 Origins of universality . . . . . . . . . . . . . . . . . . 18410.2 Universality and an operator-valued extended Voronin theorem190

10.2.1 A first quantum analog of the universality theorem . . 19010.2.2 A more general version of quantized universality . . . 194

11 Concluding Comments and Future Research Directions 197

A Riemann’s Explicit Formula 203

B Natural Boundary Conditions for ∂c 207

C The Momentum Operator and Normality of ∂c 211

D Finiteness of Some Useful Integrals 219

E The Spectral Mapping Theorem 223

F The Range and Growth of ζ(s) on Vertical Lines 235F.0.1 Estimates for the modulus of ζ(s) along vertical lines

within the critical strip . . . . . . . . . . . . . . . . . 236F.0.2 The range of ζ(s) and its Euler factors to the right of

the critical strip . . . . . . . . . . . . . . . . . . . . . 241

G Some Extensions of the Universality of ζ(s) 245G.0.1 A first extension of Voronin’s original theorem . . . . 245G.0.2 Further extensions to L-functions . . . . . . . . . . . . 247

Acknowledgements 251

Author Index 277

Subject Index 279

Index of Symbols 281

Conventions 283

ix

x

Chapter 1

Introduction

The theory of fractal strings and their complex dimensions investigatesthe geometric, spectral and physical properties of fractals and precisely de-scribes the oscillations in the geometry and the spectrum of fractal strings;see, in particular, [Lap-vF2–4]. Such oscillations are encoded in the complexdimensions of the fractal string, which are defined as the poles of the corre-sponding geometric zeta function.This theory has a variety of applicationsto number theory, arithmetic geometry, spectral geometry, fractal geometry,dynamical systems, geometric measure theory, mathematical physics andnoncommutative geometry; see, for example, [Lap2–3, Lap-vF1–4, Lap7];see, in particular, Chapter 13 of the second edition of [Lap-vF4] for a surveyof some of the recent developments in the theory.

The spectral operator was ‘heuristically’ introduced by M.L. Lapidusand M. van Frankenhuijsen in their theory of complex dimensions in fractalgeometry and number theory [Lap-vF3, Lap-vF4]. It is a map that sendsthe geometry of generalized fractal strings onto their spectrum.The corre-sponding inverse spectral problem was first considered by M.L. Lapidus andH.Maier in their work [LapMa2] on a spectral reformulation of the Riemannhypothesis in connection with the question “Can One Hear The Shape of aFractal String?”(See also [LapMa1].) The spectral operator (denoted by a)is defined on a suitable Hilbert space as the operator mapping the count-ing function of a generalized fractal string η to the counting function ofits associated spectral measure, ν = η ∗ h, where∗ denotes the multiplicativeconvolution of measures and h is the harmonic string (see Section 2.2),

Nη(x) 7−→ Nν(x)

1

or equivalently, and under the change of variable x = et,

f(t) 7−→ a(f)(t) =∞∑

n=1

f(t− log n) ,

where f is the counting function of the underlying generalized fractal string.It relates the spectrum of a fractal string with its geometry. The spectraloperator also has an operator-valued Euler product representation, whichprovides a counterpart to the usual Euler product expansion for the Riemannzeta function ζ = ζ(s), but is conjectured in [Lap-vF3, Lap-vF4] to be alsoconvergent (in a suitable sense) in the critical strip {0 < Re(s) < 1} of thecomplex plane:

a(f)(t) = ζ(∂)(f)(t) =∏

p∈P

(1− p−∂)−1(f)(t) with ∂ := ddt ,

where P denotes the set of prime numbers and for each p ∈ P, the operator-valued pth Euler factor is defined by

ap(f)(t) = ζp(∂)(f)(t) = (1− p−∂)−1(f)(t),

with ζp(s) := (1 − p−s)−1.We will show, in particular, that ∂ (to be alsosometimes denoted by ∂c later on) is an unbounded normal operator: ∂∗∂ =∂∂∗, where ∂∗ denotes the adjoint of ∂ and is given by ∂∗ = 2c−∂. Intuitively,∂ can be thought of as being the ‘infinitesimal shift’ of the real line. In fact, itwill be shown to generate the natural translation group on the function spaceHc (see Section 6.3).We note that the last two equalities are understood inthe sense of the functional calculus for unbounded normal operators.Morespecifically, ∂ denotes here a suitable realization of the differential operatorddt , acting on an appropriate Hilbert space Hc, depending on a parameterc ∈ R.

Here and thereafter, we write interchangeably

{Re(s) = α} or {s ∈ C : Re(s) = α}

in order to denote the vertical line of abscissa α ∈ R, and similarly, we let(for α ∈ R ∪ {±∞})

{Re(s) > α} := {s ∈ C : Re(s) > α}

2

denote the corresponding open right half-plane (by convention, for α = +∞,it is empty whereas for α = −∞, it is all of C.) Furthermore, for −∞ < α ≤β < +∞, we denote the corresponding vertical strip by

{α < Re(s) < β} := {s ∈ C : α < Re(s) < β}(and similarily in the case of large instead of strict inequalities). Finally,for example, the critical strip {0 < Re(s) < 1} may also be denoted0 < Re(s) < 1.

The goal of this book is to study the spectral operator a = ac, de-fined in our framework as a quantized Dirichlet series, its operator-valuedtruncations and Euler product. In this work we provide a rigorous analyticfunctional framework enabling to study the spectral operator.We describethe spectra of the infinitesimal shifts ∂c, the spectral operator and theirtruncations.We introduce two new modes of invertibility for ac; namely:almost invertibility and quasi-invertibility which turn out to be naturallyrequired to obtain a necessary and sufficient condition on its invertibility,thereby enabling us to answer the following question:“What kind of geomet-ric information about a fractal string of dimension c ∈ (0, 1) can be derivedfrom its vibrational spectrum?”.We show that an answer to this question isintimately related to the location of the critical zeros of the Riemann zetafunction and as a result, we obtain a spectral reformulation of the Riemannhypothesis (RH).

Moreover, we show the existence of several mathematical phase transi-tions for ac at the critical fractal dimensions c = 1

2 and c = 1 that are relatedto the shape of its spectrum, its boundedness, its invertibility and quasi-invertibility.We also obtain a ‘natural’ quantization (or operator-valued ver-sion) of Voronin’s theorem of the universality of the Riemann zeta functionand some of its generalizations. In our proposed version of the quantizationof the universality of ζ(s), the role played by the complex variable s inthe classical universality theorem is now played by the family of ‘truncatedinfinitesimal shifts’. Furthermore, we derive a condition ensuring the quasi-invertibility of the spectral operator in the left-hand side of the critical strip{0 < Re(s) < 1

2} (i.e., for 0 < c < 12 ) which is related to the universality

of ζ(s) on the right-hand side of the critical strip {12 < Re(s) < 1} (i.e., for

12 < c < 1).

We close this introduction by providing a relatively detailed descrip-tion of the contents of the remainder of this book (Chapters 2–10, along

3

with Appendices A–G):

In Chapter 2, we recall the definition of both classes of ordinary andgeneralized fractal stings. The latter can be viewed as natural generaliza-tions of the measures associated to ordinary fractal strings. Furthermore,wepresent some fundamental tools that are needed in our study of the spec-tral operator.We also recall the explicit formulas for generalized fractalstrings; such formulas, which generalize Riemann’s original explicit formula(see, e. g., [Edw, Ing, Pat]), were obtained and applied in the context offractal geometry, by M. L. Lapidus and M. van. Frankenhuijsen in [Lap-vF1–4]. The explicit distributional formula provides a representation of η, in thedistributional sense, as a sum over its complex dimensions (defined as thepoles of an associated geometric zeta function) which encode in their realand imaginary parts important information about the (geometric, spectralor dynamical) oscillations of the underlying fractal object. From a math-ematical and historical perspective, it is noteworthy that the original ex-plicit formula was first obtained by Riemann in 1858 as an analytical toolto understand the distribution of the primes. It was later extended by vonMangoldt and led in 1896 to the first rigorous proof of the Prime Num-ber Theorem, independently by Hadamard and de la Vallee Poussin. (See[Edw, Ing, Ivi, Pat, Tit].) In [Lap-vF3, Section 5.5], the interested readercan find a discussion of how to recover the Prime Number Theorem, alongwith a suitable form of Riemann’s original explicit formula and its variousnumber theoretic extensions (and more general results given in [Lap-vF3,Chapter 5]).

In Chapter 3, we discuss the Minkowski measurability criterion ob-tained in [LapPo2] (and announced in [LapPo1]) for fractal strings, and givean overview of the work of the second author and C.Pomerance (in [LapPo1,LapPo2]) on the direct spectral problem for fractal strings and the (mod-ified) Weyl–Berry conjecture.More specifically, the authors of the article[LapPo2] have studied the corresponding direct spectral problem for fractalstrings. They thereby have resolved in the affirmative the (one-dimensional)modifiedWeyl–Berry conjecture (as formulated in [Lap1]) according to whichif a fractal string L is Minkowski measurable of dimensionD ∈ (0, 1), then itsspectral counting function Nν(x) has a monotonic asymptotic second term,proportional to M(L)xD. Furthermore, it was shown in [LapPo1, LapPo2]that the underlying constant of proportionality is expressed in terms of thepositive constant −ζ(D), where D ∈ (0, 1), and hence, that the above directspectral problem is connected to the Riemann zeta function ζ = ζ(s) in the

4

critical interval {0 < s < 1}.

Moreover, we present some of the main results obtained in the workof the second author and H.Maier (in [LapMa2], announced in [LapMa1])on the original inverse spectral problem for fractal strings. The problem ofdeducing geometric information from the spectrum of a fractal string, orequivalently, of addressing the question

“Can one hear the shape of a fractal string?”,

was first studied by the the authors of [LapMa1, LapMa2]. The inversespectral problem they studied is the exact converse of the modified Weyl–Berry conjecture for fractal strings (from [Lap1]), as established in [LapPo1,LapPo2]. It can be stated as follows:

“Let L be a given standard fractal string whose dimension isD ∈ (0, 1). If this string has no oscillations of order D in itsspectrum, can one deduce that it is Minkowski measurable (i.e.,that it has no oscillations of order D in its geometry, accordingto the Minkowski measurability of [LapPo2])?”

More specifically, the inverse spectral problem they considered was the fol-lowing one:“Given any fixed D ∈ (0, 1), and any fractal string L of dimension D such

that for some constant cD > 0 and some δ > 0, we have

Nν(x) =W (x)− cDxD +O(xD−δ), as x→ +∞,

is it true that L is Minkowski measurable?”

The authors of [LapMa1, LapMa2] have shown that this question ala Mark Kac (but interpreted rather differently than in [Kac]) “Can onehear the shape of a fractal string?”, is intimitely connected with the Rie-mann hypothesis and thereby, with the presence of the critical zeros of theRiemann zeta function (i.e., of the zeros of ζ = ζ(s) in the critical strip{0 < Re(s) < 1}).More specifically, they proved that for a given D ∈ (0, 1),this inverse spectral problem has an affirmative answer for every fractalstring of dimension D if and only if the Riemann zeta function does nothave any zeros along the vertical line {Re(s) = D}: ζ(s) 6= 0 for all s ∈ C

such that Re(s) = D. It follows, in particular, that the inverse spectralproblem has a negative answer in the ‘midfractal case’ when D = 1

2 (be-cause ζ has a zero, and even infinitely many zeros, along the critical line

5

{Re(s) = 12}).Moreover, it follows that this inverse spectral problem has

an affirmative answer for all fractal strings whose dimension is an arbitrarynumber D ∈ (0, 1) − {1

2} if and only if the Riemann hypothesis is true.

In Chapter 4, we recall the ‘heuristic’ definition of the spectral opera-tor a = ac as introduced for the first time in [Lap-vF3, Lap-vF4], using theexplicit formulas from the just mentioned references. Following [Lap-vF3,Lap-vF4], we give two ‘heuristic’ representations of the spectral operator:an additive representation and a multiplicative one. (By ‘heuristic’ here, wemean that these representations were given in [Lap-vF3, Lap-vF4] withoutany precise functional analytic framework enabling us to study these opera-tors.)We also define the operator-valued Euler factors ap (where p ∈ P is aprime number) and Euler product

∏p∈P ap, the detailed study of which will

be the object of part of Chapter 7.

In Chapter 5, we precisely define a suitable weighted Hilbert space Hc

which depends on a parameter c. Namely, Hc = L2(R, e−2ctdt), the spaceof all complex-valued square-integrable functions with respect to the weightfunction t 7→ e−2ct, where c ∈ R is fixed.Based on the geometric part of thework of the second author and C. Pomerance in [LapPo2] (in relation withthe characterization of Minkowski measurability and of Minkowski nonde-generacy, see Section 3.1), such a space Hc can be viewed intuitively (whenc ∈ (0, 1)) as a suitable realization and extension in our context of the spaceof fractal strings of dimension D not exceeding c.

Furthermore, we define the infinitesimal shift or “differentiation opera-tor”∂ = d

dt (often denoted by ∂ = ∂c in order to emphasize the dependenceon the parameter c), acting on the Hilbert space Hc.We show that the do-main of ∂c is naturally equipped with boundary conditions which are satisfiedby the geometric counting functions of (generalized) fractal strings. Theseboundary conditions are suggested in part by the aforementioned work in[LapPo1, LapPo2].

Moreover, we prove that the infinitesimal shift ∂c is a normal, un-bounded linear operator on Hc (therefore, it commutes with its adjoint) andthat its real part is equal to c. It follows, in particular, that the associatedoperator 1

i ∂c is self-adjoint (as is the case of the standard momentum op-erator 1

i ∂0 in quantum mechanics) if and only if c = 0; see also AppendixC. (Here, we let i :=

√−1.)

6

In Chapter 6, we show that the differentiation operator ∂c is indeed theinfinitesimal shift of the real line acting on Hc, and we study the associated(shift) group of operators on Hc for any c ∈ R. Furthermore, after havingrecalled some basic definitions concerning the spectral theory of (possiblyunbounded) normal operators acting on Hilbert spaces, we give a completedescription of the spectrum of the infinitesimal shift ∂c as a suitable sub-set of the complex plane.We also exhibit the dependence of this spectrumon the asymptotic behaviour of the counting functions of fractal strings atinfinity.More specifically, we show that the spectrum of ∂c is purely con-tinuous and coincides with the vertical line {Re(s) = c}. (In particular, theinfinitesimal shift ∂c does not have any eigenvalues.) Hence, in the case ofordinary fractal strings (for which c ∈ (0, 1)), the spectrum of ∂c is a subsetof the critical strip {0 < Re(s) < 1}, a fact which is crucial throughout therest of the book.However, we note that here, the parameter c is allowed tobe any real number although for the purpose of the exposition (and since itsimplifies some of the statements), we often assume that c > 0.

Furthermore, we introduce and study a family of truncated operators,namely, the family of truncated infinitesimal shifts {∂(T )}T>0, which willplay a key role in the remainder of this book, particularly in Chapter 8 (inrelation with the notions of quasi-invertibility and almost invertibility) aswell as in Chapter 10 (in relation with the notion of quantized universal-ity). In some specific sense, in those two chapters (as well as in Section 7.7),the family of truncated infinitesimal shifts will be an approximate substitutefor the complex variable s when quantizing the Riemann zeta function orother related arithmetic zeta functions.

Moreover, if we truncate ∂c appropriately (by using the functional cal-culus and a corresponding form of the spectral mapping theorem for un-bounded normal operators described in Appendix E), and avoid the valuec = 1 corresponding to the pole of ζ(s) at s = 1, then for each given T > 0,

the associated truncated infinitesimal shift ∂(T )c is shown to have for spec-

trum the vertical line segment [c− iT, c+ iT ].

In Chapter 7, we precisely define the spectral operator a = ac. Againusing the functional calculus for unbounded normal operators, we let ac :=ζ(∂c) (or a := ζ(∂), in short), where ζ = ζ(s) is the classic (complex-valued)Riemann zeta function. (See Section 7.2.) In this precise sense, the spec-tral operator can therefore be viewed as a suitable version of a quantized

7

Riemann zeta function. (Note that in the corresponding quantization proce-dure, the infinitesimal shift ∂ = ∂c is substituted for the complex variables.) It follows from the functional calculus form of the spectral theorem forunbounded normal operators [Ru2] that the spectral operator a = ac is a(possibly unbounded) normal operator. Its spectrum will be determined inChapter 10, where it will also be shown that ac is bounded for c > 1 andunbounded for c ≤ 1.

Moreover, we show (in Section 7.1 and in Section 7.2, respectively) thatfor every c > 1, the spectral operator ac coincides with the correspondingoperator-valued (or quantized) Dirichlet series

∑∞n=1 n

−∂c and Euler prod-uct

∏p∈P(1−p−∂c)−1, where (as before) P denotes the set of all primes.This

provides a first justification of the rigorous definition of a = ac and of thesemi-heuristic formulas (from [Lap-vF3, Lap-vF4]) discussed in Chapter 4in connection with the heuristic spectral operator. Here, the above infiniteseries and infinite product converge in B(Hc), the Banach algebra (even C∗-algebra) of bounded linear operators on the Hilbert space Hc.

A central theme of this chapter (i.e., Chapter 7) is the study of theanalogies and the differences between the above quantized Riemann zetafunction (i.e., the spectral operator a = ac), the quantized Dirichlet seriesand the quantized Euler product, and their complex-valued counterparts,the classic Riemann zeta function ζ = ζ(s) (or its global analog, ξ = ξ(s),the completed Riemann zeta function), Dirichlet series

∑∞n=1 n

−s and Eulerpoduct

∏p∈P(1 − p−s)−1.We examine three cases: c > 1, 0 < c < 1, and

c ∈ R, depending on the situation under consideration.

In particular, by using an analytic continuation of ζ(s), valid forRe(s) >0 or for all s ∈ C, respectively, we further justify the definition of ac forc > 0. In doing so, we obtain a quantum analog of the classic analytic con-tinuation of ζ(s) in the half-plane {Re(s) > 0} and in the whole complexplane C (i.e., we obtain ‘analytic continuation’ representations of ac for allc > 0 and for all c ∈ R, respectively). In the process, we obtain and study anatural quantum (i.e., operator-valued) analog Ac = ξ(∂c) of the completed(or global) Riemann zeta function ξ(s) := π−

s2Γ( s2)ζ(s) and the associated

functional equation, ξ(s) = ξ(1 − s), valid for all s ∈ C. By using thefunctional calculus for linear unbounded normal operators, we show that forevery c > 1 and T > 0, the quantized Dirichlet series (for the truncatedspectral operator) along with the associated operator-valued Euler product

8

both coincide with the truncated spectral operator a(T )c .We thereby give a

‘quantized analytic continuation’ for the Dirichlet series and the associatedEuler product for the truncated spectral operator, which is valid for all c ∈ R

(and, in particular, for all c > 0). (See Sections 7.6 and 7.7.)

We also show (in Section 7.5.1) that for every c > 0 (and, in particular,for every c in the critical interval (0, 1)), the quantized analytic continua-tion,1 Dirichlet series and Euler product are well defined and coincide withone another and with the spectral operator ac, when applied to all functionsin a suitable dense subspace of the underlying Hilbert space Hc. In somesense, this shows that the usual complex-valued Dirichlet series and Eulerproduct of the (scalar-valued) Riemann zeta function, which are well knownto be divergent within the critical strip {0 < Re(s) < 1}, have a naturaloperator-valued (or quantized) counterpart, which also coincides with thespectral operator ac, at least on the above dense subspace of D(ac) (thedomain of ac) and of Hc. (Furthermore, this dense subspace is shown to bean operator core for the infinitesimal shift ∂c, that is, to be dense in the op-erator Hilbert norm of ∂c.) We point out that this result essentially provesand significantly extends a conjecture made in Subsection 6.3.2 of [Lap-vF3]and of [Lap-vF4].

In addition, we conjecture (in Subsection 7.5.2) that the quantizedDirichlet series and Euler product for the spectral operator itself ac (insteadof for its truncations {ac}T>0), now viewed as unbounded operators on Hc

when c ∈ (0, 1), admit a natural analytic continuation, valid for all c > 0(and, in particular, for all parameters c in the critical interval (0, 1)). If true,this conjecture would have important consequences for the theory developedin this book.

More specifically, we conjecture (in Subsection 7.5.2) that the afore-mentioned dense subspace of D(ac) and of Hc is also dense in the domainof ac, D(ac), equipped with the stronger operator Hilbert norm (i.e., is anoperator core for ac) and therefore determines the spectral operator ac (andthe associated operator-valued analytic continuation) uniquely. Accordingly,the spectral operator ac would be the unique (necessarily unbounded) nor-mal extension of the quantized Dirichlet series and Euler product (viewedas unbounded operators on Hc), for all c in the critical interval (0, 1). (As

1This analytic continuation is directly defined and no longer defined in terms of thetruncations of ac.

9

we already mentioned, these operators exist as bounded operators on Hc

and coincide for all c > 1, as is shown in Section 7.3 and Section 7.4.) Stillin Subsection 7.5.2, we obtain some partial result towards this conjecture,since, as was alluded to above, we show that the subspace in question is anoperator core for ∂c.

In closing the description of this chapter, we point out that in theconcrete and central case of the Riemann zeta function (viewed as the zetafunction of the field of rational numbers Q and of the associated ring ofintegers Z), Chapter 7 offers a glimpse of how the subject of quantized num-ber theory can be further developed (via essentially the same methods) forall the arithmetic zeta functions (or L-functions) occurring in number the-ory and arithmetic geometry. Indeed, Dirichlet series, Euler products andanalytic continuations of L-functions (or arithmetic zeta functions) lie atthe core of classic (analytic and algebraic) number theory. (See, for exam-ple, [ParSh1, ParSh2, HardWr, Ser, Pos, Sarn, Hid, Pat, Wei1, Wei2, Den],[Lap-vF4, esp. Appendix A] and [Lap7, esp. Appendices B, C & E], alongwith the many references therein.) Naturally, other chapters in this bookeither lay the foundations of this subject by precisely formulating the an-alytic framework within which the infinitesimal shifts ∂c and their trunca-

tions {∂(T )c }T>0 (along with the associated spectral operators ac and their

truncations {a(T )c }T>0) can be defined and studied, as well as their spectradetermined (as in Chapters 5 and 6, along with Sections 7.1, 8.1 and 9.1). Inparticular, the theory can be applied to reformulate the analog of the Rie-mann hypothesis (often called the Extended Riemann Hypothesis (ERH)in the case of general L-functions, (see, e.g., [Sarn, Lap7]) in terms of theinvertibility or the quasi-invertibility of the corresponding spectral operator(much as in Chapter 8 or in Section 9.4, respectively) and (for suitable L-functions) to provide a quantum analog of the classic notion of universality(much as in Chapter 10).

In this book, we will mainly limit ourselves to the emblematic caseof the Riemann zeta function. It is clear, however, that at least in principle(and as is briefly discussed in Chapter 11), our theory can be applied to allof the other (appropriate) number theoretic zeta functions (or L-functions),therefore justifying the use of the phrase “quantized number theory” in thetitle of this book.

In Chapter 8, we begin by defining the truncated spectral operators

10

by a(T )c = ζ(∂

(T )c ) for every c ∈ R and T > 0, via the functional calculus

for normal operators. (Note that the resulting family of truncated spectral

operators {a(T )c }T>0 provides a new quantization of the Riemann zeta func-tion, different from the one given in Chapter 7 by the spectral operatora = ζ(∂). By using the spectral mapping theorem (discussed in AppendixE), we deduce from the results obtained in Chapter 6 (about the spectrum

of the truncated infinitesimal shift ∂(T )c ) that for all c 6= 1, the spectrum of

the bounded normal operator a(T )c is equal to the range of ζ = ζ(s) along

the vertical line segment [c− iT, c+ iT ].

We next introduce the new notion of quasi-invertibility of the spectraloperator. Accordingly, the spectral operator ac is said to be quasi-invertible

if the truncated spectral operators a(T )c are invertible for every T > 0. (Note

that for c 6= 1, the latter invertibility can be understood in the usual senseof bounded operators.)

We then explain how the above result concerning the spectra of the

truncated spectral operators a(T )c (obtained in Section 6.4) can be used to

provide necessary and sufficient conditions for the invertibility (in a suitablesense) of the spectral operator (in the critical strip, that is, for c ∈ (0, 1)) andtherefore obtain a new spectral and operator theoretic reformulation of theRiemann hypothesis.More specifically, we show that the spectral operatorac is quasi-invertible (in the above sense) on the space of counting func-tions of generalized fractal strings with Minkowski dimension not exceedingc ∈ (0, 1) (that is, more mathematically, ac is quasi-invertible on Hc) if andonly if ζ(s) does not have any zeros on the vertical line {Re(s) = D}. Inother words, the truncated spectral operator a

(T )c = ζ(∂

(T )c ) is invertible for

every T > 0 if and only if ζ(s) does not have any zeros on the vertical line{Re(s) = c}.

It follows that for every c ∈ (0, 1), the c-partial Riemann hypothesisis true if and only if the spectral operator ac is quasi-invertible. Hence, thespectral operator ac is not quasi-invertible in the midfractal case when c = 1

2 ,and it is quasi-invertible everywhere else (i.e., for all c ∈ (0, 1) with c 6= 1

2)if and only if the Riemann hypothesis is true.This result sheds new lighton the work of the second author and H.Maier [LapMa1, LapMa2] whichprovided a spectral reformulation of the Riemann hypothesis expressed interms of inverse spectral problems for the vibrations of fractal strings. Seealso the later reformulation of the results of [LapMa1, LapMa2] given in

11

[Lap-vF2–4] in terms of explicit formulas and rigorously defined complexdimensions. (See, especially, Sections 6.3.1 and 6.3.2 along with Chapter 9in [Lap-vF4].) Our work provides a precise and rigorous functional analyticjustification of this latter reformulation (see, especially, [Lap-vF4, Corollary9.6]). It also shows that the question of the quasi-invertibility of the spectraloperator (now properly formulated as a precise mathematical problem) isintimately connected with the location of the critical zeros of the Riemannzeta function, and therefore with the Riemann hypothesis.

In Chapter 9, we begin by determining (in Section 9.1) the spectrumof the spectral operator a = ac, as precisely defined in Sections 6.1 and7.2.More specifically, by means of the functional calculus and using the fullstrength of the spectral mapping theorem for unbounded normal operatorsdiscussed in Appendix E,2 we show that for any fixed c ∈ R, the spectrumσ(ac) of ac = ζ(∂c) is given by

σ(ac) = cl ({ζ(s) : s ∈ C, Re(s) = c}) , (1.1)

the closure (in C) of the range of the values of the Riemann zeta functionζ(s) along the vertical line {Re(s) = c}.3

Since the possibly unbounded operator ac is invertible if and only if0 /∈ σ(ac), we deduce from Equation (1.1) various reformulations of the in-vertibility of ac; see Section 9.2 (and later on, Section 9.4). Then, in Section9.3, we show the existence of several mathematical phase transitions for thespectral operator at the critical fractal dimensions c = 1

2 and c = 1, concern-ing the shape of the spectrum of ac, its boundedness, its invertibility as wellas its quasi-invertibility. Using, in particular, results concerning the univer-sality of the Riemann zeta function among the class of nowhere vanishingholomorphic functions, we also show that the spectral operator is invertiblefor c > 1, not invertible for 1

2 < c < 1, and conditionally (i.e., under the

2See, in particular, Theorem E.9 and Theorem E.11 of Appendix E.3When c = 1, which corresponds to the pole of ζ = ζ(s) at s = 1, the above identity

(1.1) yielding the spectrum of a1 must be interpreted by assuming that Re(s) = 1 ands 6= 1 on the right-hand side of the equation.Alternatively, we can write

σ(a1) = ζ({s ∈ C : Re(s) = 1}), (1.2)

where σ(a1) := σ(a1) ∪ {∞} denotes the extended spectrum of the unbounded normaloperator a1 and the meromorphic function ζ is viewed as a continuous function withvalues in the Riemann sphere C := C∪{∞}, so that ζ(1) = ∞. (See, respectively, the firstand second meromorphic version of the spectral mapping theorem given in Theorem E.12and Corollary E.14 of Appendix E.)

12

Riemann hypothesis), invertible for 0 < c < 12 .Moreover, we prove that

the spectrum of the spectral operator is bounded for c > 1, unbounded forc = 1, equals the entire complex plane for 1

2 < c < 1, and unbounded but,conditionally, not the whole complex plane (and, in fact does not contain 0)for 0 < c < 1

2 .We therefore deduce that four types of (mathematical) phasetransitions occur for the spectral operator at the critical values (or criticalfractal dimensions) c = 1

2 and c = 1, concerning the shape of its spectrum,its boundedness (the spectral operator is bounded for c > 1, unboundedotherwise), its invertibility (with phase transitions at c = 1 and, condition-ally, at c = 1

2), as well as its quasi-invertibility (with a phase transition atc = 1

2 if and only if the Riemann hypothesis is true).

We also discuss possible mathematical and physical interpretations ofthe phase transitions of ac; see Subsection 9.3.3. Furthermore, in Section9.4 (based on [Lap8, Lap9]) we provide an “asymmetric criterion”for theRiemann hypothesis (RH), obtained by the second author by studying theinvertibility of ac for 0 < c < 1

2 .More specifically, it is shown that the spec-tral operator ac is invertible (in the usual sense of unbounded operators) forall c ∈ (0, 12) if and only if RH is true. (This is not equivalent to the anal-ogous statement for all c ∈ (12 , 1), which is not even true for a single valueof the parameter c.)4 This asymmetric reformulation of RH was obtainedearlier by the second author in [Lap8] (see also [Lap9]), using some of theresults of the present work. It relies on the non-universality of ζ(s) in the leftcritical strip {0 < Re(s) < 1

2} (i.e., for 0 < c < 12 ) and its asymmetry comes

from the universality of ζ(s) in the right critical strip {12 < Re(s) < 1} (i.e.,

for 12 < c < 1).

The corresponding “phase transition”, occurring in the midfractal casewhen c = 1

2 if and only if RH is true, is therefore a priori of a very differentnature than the (conditional) one associated with the work of the secondauthor with H. Maier in [LapMa1, LapMa2] described in Section 3.3 orthe (conditional) phase transition occurring at c = 1

2 in conjunction withthe quasi-invertibility of the spectral operator ac (as discussed in Section9.3.1).Mathematically, however, they are shown to be equivalent. Namely,they either occur simultaneously (if RH is true) or not at all (if RH is false).

4In contrast, the spectral reformulation of RH discussed in Section 3.3 (within theframework of [LapMa2]) and the corresponding one obtained in Section 9.3.1 (in terms ofthe quasi-invertibility of ac) are symmetric, in the sense that they are valid for all c ∈ (0, 1)with c 6= 1

2, or equivalently, either for all c ∈ (0, 1

2) or for all c ∈ ( 1

2, 1).

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In Chapter 10, we use in an essential manner the rigorous study (con-ducted in Section 6.4 and in Section 8.1, respectively) of the family of

truncated infinitesimal shifts {∂(T )c }T>0 and their corresponding truncated

operator-valued spectral operators {a(T )c }T>0, in order to provide a ‘quan-tum analog’ of the universality of the Riemann zeta function obtained byS.M.Voronin [Vor1–4]. First, we recall the classical universality property ofthe Riemann zeta function and then discuss some of its applications. Roughlyspeaking, the classic universality theorem states that any nowwhere van-ishing holomorphic function in a suitable open subset of the critical strip(i.e., {1

2 < Re(s) < 1}) can be uniformly approximated by suitable verti-cal translates of ζ(s). Recall that the family of truncated infinitesimal shifts

{∂(T )c }T>0 was introduced and studied in this work (in Sections 6.4 and 8.1)in order to characterize the quasi-invertibility of the spectral operator ac. Itturns out that it is also ideally suited to obtain a quantum analog of theuniversality property of the Riemann zeta function.

More specifically, using tools from the functional calculus and our de-tailed study of the spectra of the operators involved, we show that anynowhere vanishing holomorphic function of the truncated infinitesimal shifts(in a suitable open subset of the right critical strip {1

2 < Re(s) < 1}) canbe approximated by imaginary translates of the truncated spectral opera-tors. As was alluded to earlier, it turns out that in our proposed quantizedversion of the universality theorem for ζ(s), the role played by the complexvariable s in the classic universality theorem is now played by the family of

truncated infinitesimal shifts {∂(T )c }T>0.

This latter result provides a ‘natural quantization’ of Voronin’s theo-rem (and its extensions) about the universality of the Riemann zeta func-tion.We conclude that, in some sense, arbitrarily small scaled copies of thespectral operator are encoded within itself. Therefore, we deduce that thespectral operator can emulate any type of complex behavior and that it is‘chaotic’. In the long term, the theory developed in the present work and in[HerLap1–4], along with the work in [Lap7–10], is aimed in part at providinga natural quantization of various aspects of analytic (and algebraic) numbertheory and arithmetic geometry. (See also the above discussion of Chapter 7.)

Finally, in Chapter 11, we discuss various open problems and proposeseveral directions for future research concerning, in particular, the extensionof the theory to the quantization of general L-functions, the associated Eu-

14

ler product and Dirichlet series representations, as well as the correspondingglobal (or completed) arithmetic zeta functions. These proposed extensions,along with the results established in this book, should play a key role infurther ‘naturally’ quantizing various aspects of analytic number theory andarithmetic geometry.

This work is completed by seven appendices and introductory sectionsaimed at helping the reader to acquaint himself or herself with the varioussubjects needed for the present development of ‘quantized number theory’,including especially aspects of functional analysis and operator theory, spec-tral theory, mathematical physics, analytic number theory, and universality.

In Appendix A, we recall for the non-expert reader some of the basicproperties of the Riemann zeta function.The rest of this appendix is ded-icated to give an overview of Riemann’s explicit formula and explain theunderlying ‘duality’ between the prime powers and the zeros of ζ.We alsopoint out the analogy between Riemann’s explicit formula and the explicitformulas for generalized fractal strings introduced in [Lap-vF2, Lap-vF3].

In Appendix B, we establish a useful result which enables us to show,in particular, that the boundary conditions assumed for the infinitesimalshift ∂c are natural. This result plays a key role in the proof of the normalityof the unbounded linear operator ∂c provided in Section 5.3. (See Theorem5.3.1 and its proof given in that section.)

In Appendix C, we begin by observing a direct connection betweenthe normality of the infinitesimal shift ∂c and the self-adjointness of thec-momentum operator Vc = ∂c−c

i defined on Hc = L2(R, e−2ctdt).We thenestablish the self-adjointness of Vc by showing that Vc is unitarily equiva-lent to V0, the classic momentum operator of quantum mechanics, acting onH0 = L2(R, dt), the usual space of square-integrable wave functions.We de-duce from this result (along with well-known facts concerning the spectrumof V0, based on key properties of the Fourier transform) that the spectrumof Vc is equal to the real line, from which we recover the exact form ofthe spectrum of ∂c; namely, σ(∂c) = c + iR = {Re(s) = c}. Hence, in thisappendix, we provide an alternative proof of Theorem 5.3.1 of Section 5.3(about the normality of ∂c) but also of Theorem 6.2.1 of Section 6.2 (aboutthe spectrum of ∂c).

In Appendix D, we establish the finiteness of some integrals that were

15

used in the proof of the normality of the unbounded linear operators ∂c(Theorem 6.2.1) given in Section 6.2.

In Appendix E, we recall the spectral mapping theorem for boundedself-adjoint linear operators and give a rigorous statement and proof of thespectral mapping theorems (continuous and meromorphic versions) for theclass of unbounded normal linear operators. The latter theorems play a keyrole in various places in this book, in particular in deriving the results de-scribing the shapes of the spectra of the spectral operator ac and their trun-

cations a(T )c . (See, especially, Sections 8.1, 9.1 and 10.2.)We could not find

in the litterature a statement (let alone, a proof) of the spectral mappingtheorem for unbounded normal (or even, self-adjoint) operators. Hence, thisappendix may also be of independent interest.

In Appendix F, we recall, in particular, some of the known (and someof the less well-known) results about the range of ζ on vertical lines of theform {Re(s) = c}, with c ∈ R.We pay particular attention to the threedifferent cases when c > 1, 1

2 < c < 1 and 0 < c < 12 , respectively.

In Appendix G, we give a brief overview of some of the well-knownextensions of the original universality theorem (due to S.M.Voronin) for theRiemann zeta function.We give a discrete analog of the universality (due toA.Reich) and then discuss further extensions of the universality propertiesto other classes of zeta functions, such as certain L-functions and the Hur-witz zeta function.

Finally, we close this introduction by providing some general back-ground references. For the theory of unbounded self-adjoint operators andits applications to physics, especially quantum mechanics, we refer to [Br,CouHi, JoLap, Kat, ReSi 1–2, Sc]. For the more general, but less well-known,theory of unbounded normal operators (including the powerful spectral the-orem) and some of its applications, we refer to [DunSch], [Ru2], [Kat] and[JoLap]. For distribution theory, we mention [Schw1], [Schw2] and [GeSh],while for other notions of measure theory, real and functional analysis (in-cluding the theory of absolutely continuous functions) used in this paper,see, for example [Coh, Fo, Ru2]. For references about the theory of stronglycontinuous semigroups of bounded linear operators and their applications,from different perspectives, we mention [ChLat, EnNa, HilPh, JoLap, Kat,Na, Paz, ReSi1−2, Yo]. For general references on fractal geometry, we point

16

out [Fa, Lap-vF5, Man, Mat]. For various aspects of the theory of the Rie-mann zeta function and related zeta functions, see, for example [Edw, Ing,Ivi, Lap7, Lap-vF2−4, KarVo, Pat, St, Tit].

17