Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator

Herichi, Hafedh; Lapidus, Michel L.

doi:10.1090/conm/600/11948

Cited by 9 publications

(42 citation statements)

References 70 publications

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“…Later on, it was thoroughly investigated (within a rigorous functional analytic framework) in [HerLa1] and surveyed in the papers [HerLa2,HerLa3]. In the next section, we start by introducing the class of generalized fractal strings and then define the spectral operator for fractal strings.…”

Section: Approximation By Taylor Polynomials Of ζ(S)mentioning

confidence: 99%

“…In §3.2, after having recalled the original heuristic definition of the spectral operator a (as given in ), we rigorously define a = a c as well as the infinitesimal shift ∂ = ∂ c as unbounded normal operators acting on a scale of Hilbert spaces H c parametrized by a nonnegative real number c, as was done in [HerLa1,HerLa2,HerLa3]. Finally in §3.3, we briefly discuss some of the properties of the infinitesimal shifts and of the associated translation semigroups.…”

Section: Generalized Fractal Strings and The Spectral Operatormentioning

confidence: 99%

“…11 In the special case of ordinary fractal strings, Equation (3.1.10) was first observed in [La2] and used in [La3,La4,LaPo1,LaPo2,LaMa1,LaMa2,Tep,La5,LalLa1,LalLa2,HerLa1,HerLa2,HerLa3]. Furthermore, a formula analogous to the one given in Equation (3.1.10) exists for other generalizations of ordinary fractal strings, including the class of fractal sprays (also called higher-dimensional fractal strings); see [LaPo3,La2,La3,.…”

Section: Generalized Fractal Strings and The Spectral Operatormentioning

confidence: 99%

“…(See also [HerLa2,HerLa3,HerLa4].) These truncated operators were introduced in [HerLa1] to study the invertibility of the spectral operator and obtain an operatortheoretic version of the spectral reformulation of the Riemann hypothesis (obtained by the second author and H. Maier in [LaMa1,LaMa2]) as an inverse spectral problem for fractal strings.…”

Section: Introductionmentioning

confidence: 99%

“…Therefore, we deduce that the spectral operator can emulate any type of complex behavior and that it is 'chaotic'. In the long term, the theory developed in the present paper and in [HerLa1,HerLa2,HerLa3,HerLa4], along with the work in [La5,La6], is aimed in part at providing a natural quantization of various aspects of analytic (and algebraic) number theory and arithmetic geometry.…”

Section: Introductionmentioning

confidence: 99%

See 4 more Smart Citations

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Herichi¹,

Lapidus²

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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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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Section: Approximation By Taylor Polynomials Of ζ(S)mentioning

confidence: 99%

Section: Generalized Fractal Strings and The Spectral Operatormentioning

confidence: 99%

Section: Generalized Fractal Strings and The Spectral Operatormentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

Section: Introductionmentioning

confidence: 99%

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Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Herichi¹,

Lapidus²

2014

Annales De La Faculté Des Sciences De Toulouse : Mathématiques

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Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory

Lapidus

Radunović²,

Žubrinić³

2015

Fractal Geometry and Stochastics V

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In 2009, the first author introduced a class of zeta functions, called 'distance zeta functions', which has enabled us to extend the existing theory of zeta functions of fractal strings and sprays (initiated by the first author and his collaborators in the early 1990s) to arbitrary bounded (fractal) sets in Euclidean spaces of any dimensions. A closely related tool is the class of 'tube zeta functions', defined using the tube function of a fractal set. These zeta functions exhibit deep connections with Minkowski contents and upper box (or Minkowski) dimensions, as well as, more generally, with the complex dimensions of fractal sets. In particular, the abscissa of (Lebesgue, i.e., absolute) convergence of the distance zeta function coincides with the upper box dimension of a set. We also introduce a class of transcendentally quasiperiodic sets, and describe their construction based on a sequence of carefully chosen generalized Cantor sets with two auxilliary parameters. As a result, we obtain a family of "maximally hyperfractal" compact sets and relative fractal drums (i.e., such that the associated fractal zeta functions have a singularity at every point of the critical line of convergence). Finally, we discuss the general fractal tube formulas and the Minkowski measurability criterion obtained by the authors in the context of relative fractal drums (and, in particular, of bounded subsets of R N ).Keywords. Fractal set, fractal zeta functions, distance zeta function, tube zeta function, geometric zeta function of a fractal string, Minkowski content, Minkowski measurability, upper box (or Minkowski) dimension, complex dimensions of a fractal set, holomorphic and meromorphic functions, abscissa of convergence, quasiperiodic function, quasiperiodic set, relative fractal drum, fractal tube formulas.

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Towards a Fractal Cohomology: Spectra of Polya–Hilbert Operators, Regularized Determinants and Riemann Zeros

Cobler

Lapidus

2017

Exploring the Riemann Zeta Function

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Emil Artin defined a zeta function for algebraic curves over finite fields and made a conjecture about them analogous to the famous Riemann hypothesis. This and other conjectures about these zeta functions would come to be called the Weil conjectures, which were proved by Weil in the case of curves and eventually, by Deligne in the case of varieties over finite fields. Much work was done in the search for a proof of these conjectures, including the development in algebraic geometry of a Weil cohomology theory for these varieties, which uses the Frobenius operator on a finite field. The zeta function is then expressed as a determinant, allowing the properties of the function to relate to the properties of the operator. The search for a suitable cohomology theory and associated operator to prove the Riemann hypothesis has continued to this day. In this paper we study the properties of the derivative operator D = d dz on a particular family of weighted Bergman spaces of entire functions on C. The operator D can be naturally viewed as the 'infinitesimal shift of the complex plane' since it generates the group of translations of C. Furthermore, this operator is meant to be the replacement for the Frobenius operator in the general case and is used to construct an operator associated to any given meromorphic function. With this construction, we show that for a wide class of meromorphic functions, the function can be recovered by using a regularized determinant involving the operator constructed from the meromorphic function. This is illustrated in some important special cases: rational functions, zeta functions of algebraic curves (or, more generally, varieties) over finite fields, the Riemann zeta function, and cul- minating in a quantized version of the Hadamard factorization theorem that applies to any entire function of finite order. This shows that all of the information about the given meromorphic function is encoded into the special operator we constructed. Our construction is motivated in part by work of Herichi and the second author on the infinitesimal shift of the real line (instead of the complex plane) and the associated spectral operator, as well as by earlier work and conjectures of Deninger on the role of cohomology in analytic number theory, and a conjectural 'fractal cohomology theory' envisioned in work of the second author and of Lapidus and van Frankenhuijsen on complex fractal dimensions.

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Fractal Complex Dimensions, Riemann Hypothesis and Invertibility of the Spectral Operator

Cited by 9 publications

References 70 publications

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Truncated Infinitesimal Shifts, Spectral Operators and Quantized Universality of the Riemann Zeta Function

Fractal Zeta Functions and Complex Dimensions: A General Higher-Dimensional Theory

Towards a Fractal Cohomology: Spectra of Polya–Hilbert Operators, Regularized Determinants and Riemann Zeros

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