The spectral operator was introduced by M. L. Lapidus and M. van Frankenhuijsen in their reinterpretation of the earlier work of M. L. Lapidus and H. Maier [LaMa2] on inverse spectral problems and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this survey paper, we present the rigorous functional analytic framework given by the authors in [HerLa1] and within which to study the spectral operator. Furthermore, we give a necessary and sufficient condition for the invertibility of the spectral operator (in the critical strip) and therefore obtain a new spectral and operator-theoretic reformulation of the Riemann hypothesis. More specifically, we show that the spectral operator is invertible (or equivalently, that zero does not belong to its spectrum) if and only if the Riemann zeta function ζ(s) does not have any zeroes on the vertical line Re(s) = c. Hence, it is not invertible in the mid-fractal case when c = 1 2 , and it is invertible everywhere else (i.e., for all c ∈ (0, 1) with c = 1 2 ) if and only if the Riemann hypothesis is true. We also show the existence of four types of (mathematical) phase transitions occurring for the spectral operator at the critical fractal dimension c = 1 2 and c = 1 concerning the shape of the spectrum, its boundedness, its invertibility as well as its quasi-invertibility.

Abstract. A spectral reformulation of the Riemann hypothesis was obtained in [LaMa2] by the second author and H. Maier in terms of an inverse spectral problem for fractal strings. The inverse spectral problem which they studied is related to answering the question "Can one hear the shape of a fractal drum?"and was shown in [LaMa2] to have a positive answer for fractal strings whose dimension is c ∈ (0, 1) − {

A Christophe Soulé, avec une profonde amitié et admiration,à l'occasion de ses 60 ans Abstract. Nous rappelons quelques unes des principales propriétés d'universalité de la fonction zêta de Riemann ζ(s). De plus, nous expliquons comment obtenir une quantification naturelle du théorème d'universalité de Voronin (et de ses généralizations). Notre travail est basé sur la théorie des cordes fractales développée par le deuxiéme auteur et M. van Frankenhuijsen dans . Nous utilisonségalement la théorie développée dans [HerLa1-3] par les auteurs de cet article pourétudier de façon rigoreuse l'opérateur spectral (qui relie la géométrie et le spectre des cordes fractales généralizées). Cet opérateur spectral est representée comme le composé de la fonction zêta de Riemann du 'shift infinitesimal' ∂ : a = ζ(∂). Dans le processus du quantification du théorème d'universalité de la fonction zêta de Riemann, le rôle joué par la variable s (au sens classique du théoréme d'universalité) (dans le théorème classique d'universalité) est joué par la famille des shifts infinitésimaux tronqués afin d'étudier l'opérateur spectral en lien avec la reformulation spectrale de l'hypothése de Riemann vue comme un problème spectral inverse pour les cordes fractales. Ce dernier résultat fournit une version opératorielle de la reformulation spectrale obtenue par le second auteur et H. Maier dans [LaMa2]. Notre présent travail au long terme, ainsi que [La5, La6], a en partie pour but d'obtenir une quantification naturelle de divers aspects de la théorie analytiques des nombres et de la géométrie arithmétiques.2010 Mathematics Subject Classification. Primary 11M06, 11M26, 11M41, 28A80, 32B40, 47A10, 47B25, 65N21, 81Q12, 82B27. Secondary 11M55, 28A75, 34L05, 34L20, 35P20, 47B44, 47D03, 81R40.Key words and phrases. Riemann zeta function, Riemann zeros, Riemann hypothesis, spectral reformulations, fractal strings, complex dimensions, explicit formulas, geometric and spectral zeta functions, geometric and spectral counting functions, inverse spectral problems, infinitesimal shift, truncated infinitesimal shifts, spectral operator, truncated spectral operators, universality of the Riemann zeta function, universality of the spectral operator.

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The spectral operator was introduced heuristically by M. L. Lapidus and M. van Frankenhuijsen in their reinterpretation of the earlier work of M. L. Lapidus and H. Maier [LapMa2] on inverse spectral problems for fractal strings and the Riemann hypothesis. In essence, it is a map that sends the geometry of a fractal string onto its spectrum. In this book, we provide a rigorous functional analytic framework within which to study the spectral operator. Namely, we introduce an appropriate weighted Hilbert space H c , 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., Minkowski or box) dimension of the underlying fractal strings.The spectral operator a c itself can be viewed as a suitable 'quantization' of the celebrated Riemann zeta function ζ(s). In fact, it is precisely defined here as the composite map of ζ(s) and of the infinitesimal shift (of the real line) ∂ c = d dt : a c = ζ(∂ c ), acting on the Hilbert space H c = L 2 (R, e −2ct dt).We show that the infinitesimal shift ∂ c is an unbounded normal operator on H c , with spectrum the vertical line Re(s) = c. Applying the functional calculus for unbounded normal operators combined with a suitable version of the spectral mapping theorem, we deduce that a c is well defined and that its spectrum is equal to the closure of the range of ζ(s) on the vertical line Re(s) = c. Furthermore, we show that ∂ c is indeed the infinitesimal shift of the real line acting on H c , and we study the associated (shift) group of operators on H c , 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' aFor every c > 1, we show that the spectral operator a c coincides with the corresponding operator-valued Dirichlet series ∞ n=1 n −∂c and Euler 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) in the half-plane Re(s) > 0 and in the whole complex plane C (i.e., we obtain 'analytic continuation' representations of a c for c > 0 and for c ∈ R, respectively). In the process, we obtain and study a natural quantum (i.e., operator-valued) analog A c = ξ(∂) of the completed (or global) Riemann zeta function ξ(s) = π − s 2 γ( s 2 )ζ(s) and the associated functional equation.iWe then apply this theory to obtain a necessary and sufficient condition for the invertibility (in a suitable sense) of the spectral operator (in the critical strip) and therefore obtain a new spectral and operator-theoretic reformulation of the Riemann hypothesis. More specifically, we show that the spectral 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-invertible in the midfractal case where c = 1 2 , and it is quasi-invertible everywhere...