Quantized Number Theory, Fractal Strings and the Riemann Hypothesis

The following sections are included:

• Dedication
• Overview
• Preface
• Contents
• List of Figures
• List of Tables
• Conventions

The theory of fractal strings and their complex dimensions investigates the geometric, spectral and physical properties of fractals and precisely describes the oscillations in the geometry and the spectrum of fractal strings; see, in particular, [Lap-vF2–4]. Such oscillations are encoded in the complex dimensions of a fractal string, which are defined as the poles of the corresponding geometric zeta function. This theory has a variety of applications to number theory, arithmetic geometry, spectral geometry, harmonic analysis, fractal geometry, dynamical systems, geometric measure theory, mathematical physics and noncommutative geometry; see, for example, [Lap2–3, Lap-vF1–4] and [Lap7]; see, in particular, Chapter 13 of [Lap-vF4] for a survey of some of the recent developments in the theory…

In this chapter, we begin by introducing in Section 2.1 the class of generalized fractal strings and recalling some of the fundamental tools that were provided in [Lap-vF4, Chapter 4] in order to study them. In the development of the theory of fractal strings and their complex dimensions, generalized fractal strings constitute a class of strings which, in a suitable sense, generalizes the class of one-dimensional ordinary fractal strings. Once associated to an ordinary fractal string, a generalized fractal string is viewed as a measure (see Definition 2.1.1) encoding geometric or spectral information about the fractal set; see Example 2.1.3 below and [Lap-vF4, Chapter 4], along with Remark 2.1.4…

In this chapter, we recall some of the main results on Minkowski measurability criteria and also on the modified Weyl–Berry conjecture for fractal strings obtained by the second author and C. Pomerance in [LapPo1, LapPo2], as well as some of the main results obtained in [LapMa1, LapMa2] by the second author and H. Maier in their study of the inverse spectral problem for fractal strings and its connection to a reformulation of the Riemann hypothesis…

The spectral operator, henceforth denoted by 𝔞 = 𝔞_c, was introduced by M. L. Lapidus and M. van Frankenhuijsen in their development of the theory of complex fractal dimensions; see [Lap-vF3, Subsection 6.3.2]. It maps the density of geometric states to the density of spectral states. It therefore plays an important role in relating the spectrum of a generalized fractal string to its geometry; see [Lap-vF3, Subsection 6.3.1] and then, [Lap-vF4, Section 6.3], along with the much earlier 2004 paper [Lap-vF6]…

In this chapter, we rigorously define the infinitesimal shift of the real number line, or the first-order differentiation operator, $A={\partial }_c:=\frac{d}{dt}$, along with its domain D(∂_c), which is a weighted Sobolev space, viewed as a suitable subspace of the weighted Hilbert space ℍ_c. In Section 5.1, we define the weighted Hilbert space ℍ_c as the space of complex-valued Lebesgue measurable and square-integrable functions over ℝ with respect to the nonstandard weight μ_c(dt) = e^−2ctdt. Here, c ≥ 0 (or, more generally, c ∈ ℝ) is a fixed parameter, playing the role of the underlying dimension. Furthermore, dt is the usual Lebesgue measure on ℝ…

In this chapter, we give a precise description of the spectrum of the infinitesimal shift $A:={\partial }_c=\frac{d}{dt}$, when it is defined (as in Chapter 5) on a suitable dense subspace of ℍ_c. Here, c ≥ 0 (or, more generally but equivalently, c ∈ ℝ) is arbitrary. We also define two truncated families of spectral operators, called the (unsymmetric) truncated infinitesimal shifts, and determine their spectra. Furthermore, we study the semigroup generated by ∂_c and show that it is a translation (or shift) semigroup and is, in fact, the shift group acting by translations on all of the functions belonging to ℍ_c…

In this chapter, we rigorously define the spectral operator 𝔞 = 𝔞_c and show that for every value of c ∈ ℝ, it is naturally expressed as 𝔞 = ζ(∂_c), the composite map of the Riemann zeta function with the differential operator or infinitesimal shift ∂_c, acting on a suitable domain in the weighted Hilbert space ℍ_c indexed by the fractal dimensional parameter c. (See Figure 4.1 on page 57, along with the third level of the ‘symbolic tower’ of Figure 4.2 on page 62)…

In this chapter, given T ≥ 0, we introduce the T^±-truncated spectral operators ${\mathfrak{a}}_{\pm }^{(T)}=\zeta \left({\partial }_{\pm }^{(T)}\right)$ naturally corresponding to the truncated infinitesimal shifts ${\partial }_{\pm }^{(T)}$ introduced in Section 6.4; see Figure 7.1 on page 200, along with the top level of the ‘symbolic tower’ of Figure 4.2 on page 62. We then determine the spectrum of ${\mathfrak{a}}_{\pm }^{(T)}$, using the corresponding result about the spectrum of ∂ obtained in Section 6.2; see Section 8.1. More generally, but completely analogously, given 0 ≤ T₀ ≤ T, we define in Section 8.2 the unsymmetric (T₀,T)^±-truncated spectral operators ${\mathfrak{a}}_{\pm }^{\left(T_0,T\right)}=\zeta \left({\partial }_{\pm }^{\left(T_0,T\right)}\right)$ naturally associated with the unsymmetric (T₀, T)^±-truncated infinitesimal shifts ${\partial }_{\pm }^{\left(T_0,T\right)}$ introduced in Section 6.4. (The special case when T₀ = 0 yields ${\partial }_{\pm }^{(T)}$ and ${\mathfrak{a}}_{\pm }^{(T)}$, respectively; see Figure 7.1 on page 200 and the top level of Figure 4.2 on page 62.) We also determine their spectra; see Figures 6.2 and 6.3 on pages 133 and 143 for the case of ${\partial }_{+}^{(T)}$ and ${\partial }_{+}^{\left(T_0,T\right)}$, respectively…

In Section 9.1, and using the continuous (resp., meromorphic) version of the spectral mapping theorem for unbounded normal linear operators (see Equation (9.1.1) below), depending on whether c ≠ 1 or c = 1, respectively, we show that the spectrum σ(𝔞_c) of the spectral operator 𝔞_c is equal to the closure of the range of the Riemann zeta function along the vertical line L_c = {s ∈ ℂ : Re(s) = c}. Furthermore, as is the case for any unbounded normal operator, the spectrum of the spectral operator coincides with its essential spectrum σ_e (𝔞_c); see Theorem 9.1.1, along with Section 6.1…

The functional analytic framework used to study the spectral operator 𝔞 = ζ(∂), especially, in Chapters 7–9, along with its truncations ${\mathfrak{a}}_{\pm }^{(T)}=\zeta \left({\partial }_{\pm }^{(T)}\right)$ and ${\mathfrak{a}}_{\pm }^{\left(T_0,T\right)}=\zeta \left({\partial }_{\pm }^{\left(T_0,T\right)}\right)$, especially, in Section 6.4 and Chapter 8, led us to obtain a quantum analog of the Riemann zeta function ζ(s). Throughout this quantization process, the role played by the complex number s is now played by the unbounded normal linear operator ∂, or else by its truncations, the bounded or unbounded normal linear operators ${\partial }_{\pm }^{(T)}$ and ${\partial }_{\pm }^{\left(T_0,T\right)}$, depending on the context; see Figure 7.1 on page 200, along with the top two levels of Figure 4.2 on page 62. This mathematical framework, which, as was shown in Chapters 7, 8 and 9, naturally led to the quantization of several identities satisfied by the Riemann zeta function in analytic number theory, can also be used to further quantize other important properties satisfied by the Riemann zeta function, such as its “universality”…

We close this book by indicating several future research directions suggested by the present work.

A number of additional problems can be investigated along these lines, now that we have begun to establish a dictionary between aspects of analytic number theory and operator theory, along with spectral theory. In particular, it is expected that many identities for various number-theoretic L-functions and Dirichlet series have a natural operator-theoretic counterpart. Several examples of such identities will be provided in [HerLap4] and its sequel…

The following sections are included:

• Riemann’s Explicit Formula
• Natural Boundary Conditions for ∂_c
• The Momentum Operator and Normality of ∂_c
• The Spectral Mapping Theorem
• The Range and Growth of ζ(s) on Vertical Lines
  • Estimates for the modulus of ζ(s) along vertical lines within the critical strip
  • The range of ζ(s) and its Euler factors to the right of the critical strip
• Further Extensions of the Universality of ζ(s)
  • A first extension of Voronin’s original theorem
  • Extensions to certain L-functions and other zeta functions

The following sections are included:

• Acknowledgments
• Bibliography
• Index of Symbols
• Author Index
• Subject Index