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

Lapidus, Michel L.

doi:10.1098/rsta.2014.0240

Cited by 11 publications

(22 citation statements)

References 92 publications

(282 reference statements)

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“…In [22][23][24], Hafedh Herichi and Michel Lapidus have rigorously defined and studied the spectral operator, within a proper functional analytic setting. They have also reformulated the above criterion for the Riemann hypothesis in terms of a suitable notion of invertibility of the spectral operator; see also [33,34] for a corresponding asymmetric criterion, expressed in terms of the usual notion of invertibility of the spectral operator. Furthermore, they have shown that the (operator-valued or "quantized") Euler product for the spectral operator also converges inside the critical strip 0 < ℜ(s) < 1, where all the (nontrivial) complex zeros of the Riemann zeta function reside.…”

Section: Introductionmentioning

confidence: 99%

Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings

2018

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The theory of complex dimensions describes the oscillations in the geometry (spectra and dynamics) of fractal strings. Such geometric oscillations can be seen most clearly in the explicit volume formula for the tubular neighborhoods of a p-adic fractal string Lp, expressed in terms of the underlying complex dimensions. The general fractal tube formula obtained in this paper is illustrated by several examples, including the nonarchimedean Cantor and Euler strings. Moreover, we show that the Minkowski dimension of a p-adic fractal string coincides with the abscissa of convergence of the geometric zeta function associated with the string, as well as with the asymptotic growth rate of the corresponding geometric counting function. The proof of this new result can be applied to both real and p-adic fractal strings and hence, yields a unifying explanation of a key result in the theory of complex dimensions for fractal strings, even in the archimedean (or real) case.

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Section: Introductionmentioning

confidence: 99%

Minkowski Dimension and Explicit Tube Formulas for p-Adic Fractal Strings

2018

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“…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%

“…This involved studying an operator-valued version of ζ (s), which they called a quantized zeta function. An overview of these ideas and results can be found in [26], while a detailed exposition of the theory is provided in [23].…”

Section: Fractal Cohomologymentioning

confidence: 99%

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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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“…Lapidus [9] presents his theory of fractal strings and their complex dimensions and the connections with the Riemann zeta function and the Riemann hypothesis. The so-called spectral operator can be seen as a quantization of the Riemann zeta function.…”

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confidence: 99%