Riemann’s conjecture and a fractional derivative

Méhauté, Alain Le; Kaabouchi, A. El; Nivanen, Laurent

doi:10.1016/j.camwa.2009.08.022

Cited by 5 publications

(11 citation statements)

References 8 publications

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“…The singular points {0, 1, ∞} suggest links between 1/d-exponantial and Teichmuller-Grothendick absolute Galois group. Adjoined with its categorical Kan extension τ: {τ n } obtained through the prime-decomposition n = ωτ n ∈ N and playing the role of inverse Fourier transform, the above representation shapes, -via the pair of geodesics building the semi-circle, -the functional relationship between ζ(s) and ζ(1 −s) [21][22][23].…”

Section: Zeta Function and "α-Expona Ntiation"mentioning

confidence: 99%

“…In addition, ten years after Voronin did establish his theorem, Bagchi demonstrated [33] that the validity of the Riemann hypothesis (RH) is equivalent to the verification of the universality theorem of Voronin in the particular case where the function f (s) is replaced by ζ(s), namely: ∀ε > 0, lim inf T →∞ (1/T) × mes{τ ∈ [0, T]; max|ζ(s + it) − ζ(s)| < ε} > 0. Therefore, Bagchi's inequality asserts that the nexus of RH is the self-similarity of Riemann function, the property explicitly content in its link with Z τ n α [21][22][23]. Nevertheless, since the distribution of the zeta zeros is unknown, it must be observed that the restriction concerning the absence of zeros inside the compact set K does not allow one to apply the Voronin theorem to ζ(s).…”

Section: Universality Of Zeta Riemann Functionmentioning

confidence: 99%

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From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis

Riot¹,

Méhauté²

2017

Condens. Matter Phys.

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The novelty of the Jean Pierre Badiali last scientific works stems to a quantum approach based on both (i) a return to the notion of trajectories (Feynman paths) and (ii) an irreversibility of the quantum transitions. These iconoclastic choices find again the Hilbertian and the von Neumann algebraic point of view by dealing statistics over loops. This approach confers an external thermodynamic origin to the notion of a quantum unit of time (Rovelli Connes' thermal time). This notion, basis for quantization, appears herein as a mere criterion of parting between the quantum regime and the thermodynamic regime. The purpose of this note is to unfold the content of the last five years of scientific exchanges aiming to link in a coherent scheme the Jean Pierre's choices and works, and the works of the authors of this note based on hyperbolic geodesics and the associated role of Riemann zeta functions. While these options do not unveil any contradictions, nevertheless they give birth to an intrinsic arrow of time different from the thermal time. The question of the physical meaning of Riemann hypothesis as the basis of quantum mechanics, which was at the heart of our last exchanges, is the backbone of this note. From algebraic analysis of quantum mechanics to "irreversible" Feynman paths integralDespite the unstoppable success of the technosciences based on both quantum mechanics, standard particle model and cosmological model, at least two questions must be investigated among many issues that the theories leave open [1, 2]: (i) the question of the ontological status of the time and (ii) the obsessive interrogation concerning the existence or the absence of an intrinsic "arrow of time". The origin of these questions comes from the equivocal equivalence of the status of time in any types of mechanical formalisms. For example, within Newtonian vision, the observable f can be analysed algebraically using action-integral through the Lagrangian L while Poisson brackets gives time differential representations d f /dt = {H, f }. According to Noether theorem, the energy, referred to the Hamiltonian H, is no other than the tag of a time-shift independence of physical laws, namely a compact commutativity. The statistical knowledge of the high dimensions system requires (i) the definition of a Liouville measure µ L based on the symplectic structure of the phase space and (ii) the value of the configuration distribution Z C , therefore dµ ∼ (1/Z C ) e −βH , with β = 1/k B T related to the inverse of the temperature. This point of view is discretized in quantum mechanics (QM).

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Section: Zeta Function and "α-Expona Ntiation"mentioning

confidence: 99%

Section: Universality Of Zeta Riemann Functionmentioning

confidence: 99%

From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis

Riot¹,

Méhauté²

2017

Condens. Matter Phys.

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“…the situation is the same and an artefact of algebra (90°=2x45°) allows a coming back to deterministic expression • The situation changes if we consider the whole field of definition of alpha and especially 1 2 / 1 < < α [29,30,54]. In this case the information carried by the angle at infinity leads the entanglements between object and subject.…”

Section: Figure 1 See the Text For Explanationsmentioning

confidence: 99%

“…Even if it is not a mathematical demonstration of the conjecture this analysis valid the Riemann Conjecture as a geometrical phase breaking of symmetries [29,30,53,62] but also unveils the role of the Riemannian geometry. These properties associate our analysis to non-commutative geometry theory [44].…”

Section: The Role Of Riemann Zeta Function and Of The Prime Numbersmentioning

confidence: 99%

“…This paper will focus its proposals on an analogy based on the complexity involved by zeta Riemann function in particular because we have knowledge about its degeneracy in quantum physics [15,[23][24][25][26][27][28]. According to our proposal concerning a new approach to Riemann conjecture [29][30][31] we have to start from what it is probably the keystone of the problematic of complexity: the coupling between different scales.…”

Section: Introduction To Complex Systemsmentioning

confidence: 99%

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Role of Riemann's and Goldbach's hypotheses in the behaviour of complex systems:Introduction to the concept of "Sciances"

Méhauté

Tayurskiı̆

2012

J. Phys.: Conf. Ser.

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Abstract. The authors have already established a bi univocal correspondence betweenRiemann zeta functions and dynamic processes under the control of integro-differential operator of non-integer complex order. We recall that the Riemann zeta function can then be related to hyperbolic geodesics whose angles at the boundary are determined by the real part of the power laws that define the Riemann series. It is suggested that Riemann's conjecture can be reduced to a geometrical phase transition with a reduction of the parameter of order resulting from the combination of a pair symmetries associated with a quasi-self similarity of geodetics. The well-known relationship with the set of prime numbers must be considered as the result of the local existence of stationary 'state' in the dynamics. The work is focused on the 'non stationary' behaviour of Riemann zeta function. It is shown that the main characteristic of the dynamics of complex systems may be associated to a hybridizing between a pair of states and/or processes able to give a geometrical status to the concept of the time and equally to the concept of energy. It is based on the mirror properties of complementary zeta function. It is shown also that the set of prime numbers, which controls the transitions between 'states', is the simplest form of the complexity. This analysis suggests the existence of a mathematical relationship between Riemann's and Goldbach's hypothesis. Such relationship would be the base of an extension of the principles of the science for the analysis of the complexity. According to previous proposal we name this extension that enlarges the principles of the science: sciance with 'a'.

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Complex Fractional Differential Operators Geometrical Phase Transition and Riemann Conjecture

Méhauté

Tayurskiı̆

2013

IFAC Proceedings Volumes

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Riemann’s conjecture and a fractional derivative

Cited by 5 publications

References 8 publications

From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis

From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis

Role of Riemann's and Goldbach's hypotheses in the behaviour of complex systems:Introduction to the concept of "Sciances"

Complex Fractional Differential Operators Geometrical Phase Transition and Riemann Conjecture

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