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

Herichi, Hafedh; Lapidus, Michel L.

doi:10.5802/afst.1419

Cited by 4 publications

(21 citation statements)

References 75 publications

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“…The answer to this question is affirmative, as we shall explain below; see theorem 5. By contrast, we have the following result, which was already observed in [8][9][10][11] and clearly shows that the reformulation of RH obtained in theorem 5.1 is asymmetric, in a strong sense. (a) The universality of ζ implies a much stronger result than the one used in the proof of theorem 5.2 (part (ii) of the above proof).…”

Section: Invertibility Of the Spectral Operator And An Asymmetric Refmentioning

confidence: 39%

“…For notational simplicity, we write a and b instead of a c and b c , respectively. Note that, for c ≤ 1, both a and b are unbounded linear operators (according to the results of [8][9][10][11] discussed in §4).…”

Section: Introductionmentioning

confidence: 99%

“…independently of the truth of RH), at D = 1. The first phase transition is intimately connected with the universality of the Riemann zeta function [8,11]. A brief exposition of some of the most relevant results of [8][9][10][11][12] is provided in §4.…”

Section: Introductionmentioning

confidence: 99%

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

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Section: Invertibility Of the Spectral Operator And An Asymmetric Refmentioning

confidence: 39%

Section: Introductionmentioning

confidence: 99%

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

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2015

Phil. Trans. R. Soc. A.

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“…In search of the elusive 'Frobenius operator in characteristic 0', the second author worked with H. Herichi to develop a 'Quantized Number Theory' in [23], [20], [21], [22], [26]. Here, they used an operator they denoted ∂ , which was the derivative operator on a suitable family of Hilbert spaces.…”

Section: Fractal Cohomologymentioning

confidence: 99%

“…A treatment examining the derivative operator on L 2 (R, e −2ct dt) and its use to create a 'quantized number theory' can be found in the research monograph [23], as well as in the accompanying articles [20], [21], [22] and [26].…”

Section: Derivative Operator On Weighted Bergman Spacesmentioning

confidence: 99%

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 zeta functions and complex dimensions of relative fractal drums

Lapidus

Radunović

Žubrinić

2014

J. Fixed Point Theory Appl.

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To Professor Haïm Brezis, with profound admiration, on the occasion of his 70th birthday. Avec mes sincères remerciements, pour votre amitié, vos enseignements, et votre exempleédifiant. Votreétudiant, Michel.Key words and phrases. Fractal set, fractal drum, relative fractal drum, 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, relative fractal drum, holomorphic and meromorphic functions, abscissa of convergence, quasiperiodic function, quasiperiodic set, order of quasiperiodicity, spectral asymptotics of fractal drums.* The work of Michel L. Lapidus was partially supported by the US National Science Foundation (NSF) under the research grants DMS-0707524 and DMS-1107750, as well as by the Institut des Hautes Etudes Scientifiques (IHES) in Paris/Bures-sur-Yvette, France, where the first author was a visiting professor in the Spring of 2012 while part of this work was completed.† Goran Radunović and DarkoŽubrinić express their gratitude to the Ministry of Science of the Republic of Croatia for its support, as well as to the University of Zagreb for supporting a visit of the second author to the University of California, Riverside, in the Spring of 2014 during which a part of this article was completed. Abstract. The theory of 'zeta functions of fractal strings' has been initiated by the first author in the early 1990s, and developed jointly with his collaborators during almost two decades of intensive research in numerous articles and several monographs. In 2009, the same author introduced a new class of zeta functions, called 'distance zeta functions', which since then, has enabled us to extend the existing theory of zeta functions of fractal strings and sprays to arbitrary bounded (fractal) sets in Euclidean spaces of any dimension. A natural and closely related tool for the study of distance zeta functions is the class of 'tube zeta functions', defined using the tube function of a fractal set. These three classes of zeta functions, under the name of 'fractal zeta functions', exhibit deep connections with Minkowski contents and upper box dimensions, as well as, more generally, with the complex dimensions of fractal sets. Further extensions include zeta functions of relative fractal drums, the box dimension of which can assume negative values, including minus infinity. We also survey some results concerning the existence of the meromorphic extensions of the spectral zeta functions of fractal drums, based in an essential way on earlier results of the first author on the spectral (or eigenvalue) asymptotics of fractal drums. It follows from these results that the associated spectral zeta function has a (nontrivial) meromorphic extension, and we use some of our results about fractal zeta functions to show the new fact according to which the upper bound obtained for the corresponding abscissa of meromorphic convergence is optimal. Finally...

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

Cited by 4 publications

References 75 publications

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

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

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

Fractal zeta functions and complex dimensions of relative fractal drums

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