The Riemann Zeros as Spectrum and the Riemann Hypothesis

Sierra, Germȧn

doi:10.3390/sym11040494

Cited by 23 publications

(44 citation statements)

References 82 publications

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“…Because of the unexpected association between the RH and quantum physics, and particularly the supporting evidence from random matrix theory [12][13][14] and quantum chaos [15,16], several researchers * jmcui@ustc.edu.cn † hyf@ustc.edu.cn ‡ smhan@ustc.edu.cn § cfli@ustc.edu.cn have attempted to construct a suitable quantum Hamiltonian [17][18][19][20][21][22]. The xp model is a well-known example [17,23,24], however, to our knowledge, no such Hamiltonian has been implemented in a real quantum system, e.g., more recently an operator related to the xp Hamiltonian whose eigenvalues correspond exactly to the Riemann zeros has been found [25], but unfortunately it is not hermitian and therefore it is not a properly a Hamiltonian.…”

Section: Introductionmentioning

confidence: 99%

Identifying the Riemann zeros by periodically driving a single qubit

Cui

et al. 2020

Phys. Rev. A

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The Riemann hypothesis, one of the most important open problems in pure mathematics, implies the most profound secret of prime numbers. One of the most interesting approaches to solve this hypothesis is to connect the problem with the spectrum of the physical Hamiltonian of a quantum system. However, none of the proposed quantum Hamiltonians have been experimentally feasible.Here, we report the first experiment to identify the first non-trivial zeros of the Riemann zeta function and the first two zeros of Pólya's fake zeta function, using a novel Floquet method, through properly designed periodically driving functions. According to this method, the zeros of these functions are characterized by the occurrence of crossings of quasi-energies when the dynamics of the system are frozen. The experimentally obtained zeros are in excellent agreement with their exact values. Our study provides the first experimental realization of the Riemann zeros, which may provide new insights into this fundamental mathematical problem.

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Section: Introductionmentioning

confidence: 99%

Identifying the Riemann zeros by periodically driving a single qubit

Cui

et al. 2020

Phys. Rev. A

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“…However, despite the remarkable connection with the Riemann zeros, the derivation is rather heuristic and the attempts to generalize this approach for the fluctuation term seems to be difficult using the semiclassical approach. Thus, a full quantum version of this model is considered [19], [30], but even there some connections with the Riemann zeros is lost. In this manner, taking into consideration the relations between polymer quantization and chaotic classical systems, we explore the polymer quantization of the Berry-Keating model.…”

Section: Semicalssical Approachmentioning

confidence: 99%

“…The first term corresponds to the Fourier transform of the formal eigenfunctions in the quantum H = xp model studied by Berry and Keating [13], [19]. In order to calculate the energy eigenvalues, some boundary conditions must be added in order to make the symmetric polymer operatorĤ poly a self-adjoint operator.…”

Section: The Polymer Quantum Xp-modelmentioning

confidence: 99%

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Polymeric quantum mechanics and the zeros of the Riemann zeta function

Berra─Montiel

Molgado

2018

Int. J. Geom. Methods Mod. Phys.

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We analyze the Berry-Keating model and the Sierra and Rodríguez-Laguna Hamiltonian within the polymeric quantization formalism. By using the polymer representation, we obtain for both models, the associated polymeric quantum Hamiltonians and the corresponding stationary wave functions. The self-adjointness condition provide a proper domain for the Hamiltonian operator and the energy spectrum, which turned out to be dependent on an introduced scale parameter. By performing a counting of semiclassical states, we prove that the polymer representation reproduces the smooth part of the Riemannvon Mangoldt formula, and also introduces a correction depending on the energy and the scale parameter. This may shed some light on the understanding of the fluctuation behavior of the zeros of the Riemann function from a purely quantum point of view.

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“…However, we have not yet discussed either the normalizability of the class of functions or the self-adjoint properties of the Hamiltonians in the Dirac sense which we do next. It is well known [19][20][21][22][23][24] that the class of functions x −σ+iρ is normalizable in the positive real axis Ê + in the Dirac sense (corresponding to a Dirac inner product) only for σ = 1 2 . Namely, only for the class of functions ψ ρ (…”

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confidence: 99%

Supersymmetry and the Riemann zeros on the critical line

Das

Kalauni

2019

Physics Letters B

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We propose a new way of studying the Riemann zeros on the critical line using ideas from supersymmetry. Namely, we construct a supersymmetric quantum mechanical model whose energy eigenvalues correspond to the Riemann zeta function in the strip 0 < Re s < 1 (in the complex parameter space) and show that the zeros on the critical line arise naturally from the vanishing ground state energy condition in this model. Riemann [1]generalized Euler's zeta function to the entire complex space of parameters in three essential steps. First, he extended the series representation of the zeta function to complex parameters asThis series representation can be written as a product of factors involving only prime numbers and the zeta function in (1) has an integral representation in terms of the Mellin transform of the Bose-Einstein distribution function [2]. In the second step, he expressed the zeta function in (1) in terms of the alternating zeta function asThis leads to an integral representation of the zeta function in terms of the Mellin transform of the Fermi-Dirac distribution function [2]. In this way, Riemann had defined the zeta function on the right half of the complex parameter space except for the point s = 1. In order to extend it to the left half of the plane, Riemann derived two equivalent functional relations (we give only one, the other can be obtained from this by letting s → 1 − s)This, therefore, generalizes the zeta function to the entire complex plane. The zeta function is an analytic function [3,4] which has a simple pole at s = 1 with residue 1 (the pole structure is already manifest in (2)). It vanishes for s = −2k where k ≥ 1 which can be seen from (3). These zeros are known as "trivial" zeros of the zeta function since they arise from the kinematic trigonometric factor in (3) and are the only zeros for Re s ≤ 0 (ζ(0) = − 1 2 ). Furthermore, from the representation of the zeta function in terms of products involving prime numbers, it can be shown that the zeta function has no zero for Re s > 1 (since none of the factors can vanish there). Therefore, any other "non trivial" zero of the zeta function must lie in the strip 0 < Re s < 1.(4)Riemann conjectured [1] that all other zeros of the zeta function lie on the critical line Re s = 1 2 , namely,where λ * denotes the location of a zero on the critical line. This is known as the Riemann hypothesis and so far many zeros have been calculated on the critical line numerically [5,6]. Note that the Riemann hypothesis specifies only the real part of the parameter s at a zero. It does not say anything about the location (imaginary part of the parameter s) on the critical line. The numerical calculations do not yet show any particular recurrence relation or regularity in the locations of the zeros. There is a fascinatingly beautiful symmetry in physics known as supersymmetry [7][8][9][10]. It has a simple manifestation in one dimensional supersymmetric quantum mechanics [11][12][13][14] where there are two conserved charges, Q, Q † satisfying the graded algebra (t...

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The Riemann Zeros as Spectrum and the Riemann Hypothesis

Cited by 23 publications

References 82 publications

Identifying the Riemann zeros by periodically driving a single qubit

Identifying the Riemann zeros by periodically driving a single qubit

Polymeric quantum mechanics and the zeros of the Riemann zeta function

Supersymmetry and the Riemann zeros on the critical line

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