Self-similar p-adic fractal strings and their complex dimensions

Lapidus, Michel L.; Lũ, Hùng

doi:10.1134/s2070046609020083

Cited by 13 publications

(38 citation statements)

References 21 publications

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“…Many applications and extensions of fractal string theory and/or of the corresponding theory of complex fractal dimensions can be found throughout the books La5] and in [La1,La2,La3,La4,LaPo1,LaPo2,LaPo3,LaMa1,LaMa2,HeLa,HamLa,Tep,LaPe,LaPeWi,LaLeRo,ElLaMaRo,LaLu1,LaLu2,LalLa1,LalLa2,LaRaZu,HerLa1,HerLa2,HerLa3,HerLa4,La6]. These include, in particular, applications to various aspects of number theory and arithmetic geometry, dynamical systems, spectral geometry, geometric measure theory, noncommutative geometry, mathematical physics and nonarchimedean analysis.…”

Section: Generalized Fractal Strings and The Spectral Operatormentioning

confidence: 99%

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: Generalized Fractal Strings and The Spectral Operatormentioning

confidence: 99%

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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“…In [9,10], we prove that D is the Minkowski dimension of CS 3 ⊂ Z 3 . Clearly, D is also the abscissa of convergence of the Dirichlet series defining ζ CS3 .…”

Section: Nonarchimedean Cantor Stringmentioning

confidence: 99%

“…The above theorem shows that G = 1 + 3Z 3 is the generator of the nonarchimedean Cantor string. This is a particular case of a more general construction of self-similar p-adic fractal strings [9,10]. Moreover, CS 3 is not Minkowski measurable as a subset of Q 3 .…”

Section: Nonarchimedean Cantor Stringmentioning

confidence: 99%

“…In a forthcoming paper [9], we will develop a general framework for formulating a theory of self-similar p-adic (or nonarchimedean) strings and their complex fractal dimensions. Besides answering a natural mathematical question, these results may be useful in various aspects of mathematical physics, including p-adic quantum mechanics and string theory, where extensions from the archimedean to the nonarchimedean setting have been extensively explored [12].…”

Section: Introductionmentioning

confidence: 99%

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Nonarchimedean Cantor set and string

Lapidus

Lũ

2008

J. fixed point theory appl.

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Abstract. We construct a nonarchimedean (or p-adic) analogue of the classical ternary Cantor set C. In particular, we show that this nonarchimedean Cantor set C3 is self-similar. Furthermore, we characterize C3 as the subset of 3-adic integers whose elements contain only 0's and 2's in their 3-adic expansions and prove that C3 is naturally homeomorphic to C. Finally, from the point of view of the theory of fractal strings and their complex fractal dimensions [7,8], the corresponding nonarchimedean Cantor string resembles the standard archimedean (or real ) Cantor string perfectly.

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“…We note that recent developments in the theory are described in [7, ch. 13], including a first attempt at a higher dimensional theory of complex dimensions for the special case of fractal sprays (in the sense of [30]) and self-similar tilings (see [7, §13.1], based on [63][64][65][66]72]), p-adic fractal strings and associated non-Archimedean fractal tube formulae (see [7, §13.2], based on [56][57][58][59][60]), multi-fractal zeta functions and their 'tapestries' of complex dimensions (see [7, §13.3], based on [50,55,67]), random fractal strings (such as stochastically self-similar strings and the zero-set of Brownian motion) and their spectra (see [7, §13.4], based on [51]), as well as a new approach to the RH based on a conjectural Riemann flow of fractal membranes (i.e. quantized fractal strings) and correspondingly flows of zeta functions (or 'partition functions') and of the associated zeros (see [7, §13.5], which gives a brief overview of the aforementioned book [20], In search of the Riemann zeros).…”

Section: Remark 21mentioning

confidence: 99%

Self-similar p-adic fractal strings and their complex dimensions

Cited by 13 publications

References 21 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

Nonarchimedean Cantor set and string

Towards quantized number theory: spectral operators and an asymmetric criterion for the Riemann hypothesis

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