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

Herichi, Hafedh; Lapidus, Michel L.

doi:10.48550/arxiv.1210.0882

Cited by 1 publication

(13 citation statements)

References 0 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…[ . 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 this section, we first recall (in §3.1) the notion of generalized fractal string introduced and used extensively in , for example, in order to obtain general explicit formulas applicable to various aspects of number theory, fractal geometry, dynamical systems and spectral geometry. In §3.2, after having recalled the original heuristic definition of the spectral operator a (as given in [La-vF3, La-vF4]), 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: Approximation By Taylor Polynomials Of ζ(S)mentioning

confidence: 99%

“…10 The Möbius function µ(j) equals 1 if j is a square-free positive integer with an even number of prime factors, -1 if j is a square-free positive integer with an odd number of prime factors, and 0 if j is not square-free. 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: Approximation By Taylor Polynomials Of ζ(S)mentioning

confidence: 99%

“…In this paper, we discuss the properties of the family of 'truncated infinitesimal shifts' of the real line and of the associated 'truncated spectral operators'. (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

Self Cite

View full text Add to dashboard Cite

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.

show abstract

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

confidence: 99%

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

confidence: 99%