quantum mechanics
the nontrivial Riemann zeta zeros interpreted as a spectrum of energy levels


"The Riemann zeta function is of interest to pure mathematicians because of its connection with prime numbers, but it is also a hugely important tool in quantum chaos because many calculations involving the Riemann zeta function mirror the most fundamental manipulations in semiclassical work, those concerning the energy eigenvalues of semiclassical systems and the action of the periodic orbits of those systems. Whereas the semiclassical calculations involve sums over periodic orbits of the system in question...the Riemann zeta function version contains sums over prime numbers. As much knowledge has built up about prime numbers over the years, the Riemann zeta calculations are often more tractable than the periodic orbit ones, and so can provide insight as to how the semiclassical calculations ought to proceed."

from N. Snaith's Ph.D. thesis "Random matrix theory and zeta functions" (University of Bristol, 2000)

"There's been an explosion of activity in this field - the progress in the last half dozen years because of this marriage of these two fields has been absolutely incredible."

S. Gonek, quoted by K. Sabbagh in Dr. Riemann's Zeros (Atlantic, 2002), p. 148

"Although the Riemann zeta-function is an analytic function with [a] deceptively simple definition, it keeps bouncing around almost randomly without settling down to some regular asymptotic pattern. The Riemann zeta-function displays the essence of chaos in quantum mechanics, analytically smooth, and yet seemingly unpredictable."

M.C. Gutzwiller, Chaos in Classical and Quantum Mechanics (Springer, 1990), p. 377

"One idea for proving the Riemann hypothesis is to give a spectral interpretation of the zeros. That is, if the zeros can be interpreted as the eigenvalues of 1/2 + iT, where T is a Hermitian operator on some Hilbert space, then since the eigenvalues of a Hermitian operator are real, the Riemann hypothesis follows. This idea was originally put forth by Pólya and Hilbert, and serious support for this idea was found in the resemblance between the "explicit formulae" of prime number theory, which go back to Riemann and von Mangoldt, but which were formalized as a duality principle by Weil, on the one hand, and the Selberg trace formula on the other.

The best evidence for the spectral interpretation comes from the theory of the Gaussian Unitary Ensemble (GUE), which show that the local behavior of the zeros mimics that of a random Hamiltonian. The link gives a more extended discussion of this topic."

"Gutzwiller gave a trace formula in the setting of quantum chaos which relates the classical and quantum mechanical pictures. Given a chaotic (classical) dynamical system, there will exist a dense set of periodic orbits, and one side of the trace formula will be a sum over the lengths of these orbits. On the other side will be a sum over the eigenvalues of the Hamiltonian in the quantum-mechanical analog of the given classical dynamical system.

This setup resembles the explicit formulas of prime number theory. In this analogy, the lengths of the prime periodic orbits play the role of the rational primes, while the eigenvalues of the Hamiltonian play the role of the zeros of the zeta function. Based on this analogy and pearls mined from Odlyzko's numerical evidence, Sir Michael Berry proposes that there exists a classical dynamical system, asymmetric with respect to time reversal, the lengths of whose periodic orbits correspond to the rational primes, and whose quantum-mechanical analog has a Hamiltonian with zeros equal to the imaginary parts of the nontrivial zeros of the zeta function. The search for such a dynamical system is one approach to proving the Riemann hypothesis."   (Daniel Bump)

"When this conjecture was formulated about 80 years ago, it was apparently no more than an inspired guess. Neither Hilbert nor Pólya specified what operator or even what space would be involved in this correspondence. Today, however, that guess is increasingly regarded as wonderfuly inspired, and many researchers feel that the most promising approach to proving the Riemann Hypothesis is through proving the Hilbert-Pólya conjecture. Their confidence is bolstered by several developments subsequent to Hilbert's and Pólya's formulation of their conjecture. There are very suggestive analogies with Selberg zeta functions. There is also the extensive research stimulated by Hugh Montgomery's work on the pair-correlation conjecture for zeros of the zeta function. Montgomery's results led to the conjecture that zeta zeros behave asynptotically like eigenvalues of large random matrices from the GUE ensemble that has been studied extensively by mathematical physicists...Although this conjecture is very speculative, the empirical evidence is overwhelmingly in its favor."

A.M. Odlyzko from "The 1022-nd zero of the Riemann zeta function".

"We have all this evidence that the Riemann zeros are vibrations, but we don't know what's doing the vibrating."

M. du Sautoy, The Music of the Primes (Fourth Estate, 2003) p.280

a couple of historical notes:

Andrew Odlyko's correspondence with George Pólya concerning the origins of the 'Hilbert-Pólya conjecture'.

M. du Sautoy describes Jon Keating's discovery in Riemann's notes of possible evidence that he may have considered this approach to his Hypothesis long before Hilbert or Pólya.

popular/general introductions

Michael Berry's research

partially successful attempts to produce the required operator/dynamics

other related material

quantum mechanics and number theory (other contexts)

quantum chaos links (workgroups, introductory articles, etc.)

random matrix theory and the Riemann zeta function


popular/general introductions

B. Cipra, "A Prime Case of Chaos" (An excellent introduction to Berry's conjecture, etc. for general readership.)

E. Klarreich, "Prime Time", New Scientist, 11/11/00 (another popular exposition)

A. Granville, "Prime possibilities and quantum chaos" (from MSRI Emissary, spring 2002)

"The Mark of Zeta" - introductory essay on the Riemann Hypothesis and Riemann's zeta function (I. Peterson)

"The Return of Zeta" - sequel article by I. Peterson on links between the Riemann Hypothesis, random matrix theory and quantum chaos

B. Hayes, "The spectrum of Riemannium", American Scientist, July-August 2003

This is a popular article on the fortuitous meeting between Dyson and Montgomery in 1972, the connection between quantum chaos, random matrix theory and the Riemann zeta function, etc.

J. Keating, "Physics and the Queen of Mathematics", Physics World, April 1990, p.46

"...number theory, once considered by mathematicians to be a field with no application to the other sciences, is now proving to be of considerable use to physicists, both as a working tool and as a guide to possible directions of future research. It may even be the case that physics can help in the solution of some of the important problems in this, one of the purest areas of pure mathematics."

B. Cipra, "Prime formula weds number theory and quantum mechanics", Science 274 (1996) 2014.

G. Sierra, "A physics pathway to the Riemann hypothesis" (preprint 12/2010)

[abstract:] "We present a brief review of the spectral approach to the Riemann hypothesis, according to which the imaginary part of the non trivial zeros of the zeta function are the eigenvalues of the Hamiltonian of a quantum mechanical system."

W. Blum, "Wird dank der Quantenphysik die Riemannsche Vermutung endlich bewiesen?" (an article in German from the newspaper Die Zeit on the connection between quantum chaos and the zeros of the Riemann zeta function)

WWN notes - "Physics and the zeros of the zeta-function" (part of a work-in-progress)

N. Patson, "Review of quantum chaology and structural complexity approaches to characterising global behaviour with application to primes" (A more mathematically advanced introduction to these issues.)

Z. Rudnick, "Zeta functions in arithmetic and their spectral statistics", Proceedings of a special semester at the Institut Poincaré, 1996.

[from introduction:] "The Riemann zeta function $\zeta(s)$ serves as an important model in many investigations into the theory of Quantum Chaos. My aims in these lectures, which are directed at physicists, are to explain some of the basic properties of $\zeta(s)$ used by number theorists, and discuss the spectral statistics of their zeros in connection with Random Matrix Theory."

T. Kriecherbauer, J. Marklof and A. Soshnikov, "Random matrices and quantum chaos" (brief introductory article, including a description of how this applies to the Riemann Hypothesis)

H.-J. Stöckmann, Quantum Chaos: An Introduction (Cambridge University Press, 1999)


Michael Berry's research

The following is an excellent survey article on the work of Berry and Keating in this area:

M.V. Berry and J.P. Keating "The Riemann zeros and eigenvalue asymptotics", SIAM Review, 41, No. 2 (1999) 236-266.

See also:

M.V. Berry and J.P. Keating, "A compact hamiltonian with the same asymptotic mean spectral density as the Riemann zeros", J. Phys. A 44 (2011) 285203 (14pp)

M.V. Berry, "Quantum chaology, prime numbers and Riemann's zeta function" (paper presented at International Conference on Nuclear and Particle Physics, Glasgow, 1993)

M.V. Berry, "Riemann's zeta function: a model for quantum chaos?"

M.V. Berry and J.P. Keating, "H = xp and the Riemann zeros", from Supersymmetry and Trace Formulae: Chaos and Disorder, ed. Lerner, et. al. (Kluwer/Plenum, 1999).

M.V. Berry, "Semiclassical formula for the number variance of the Riemann zeros", Nonlinearity 1 (1988) 399-407.

M.V. Berry and J.P. Keating, "A new asymptotic representation for zeta(1/2 + it) and quantum spectral invariants", Proceedings of the Royal Society fo London A 437 151-173.

M.V. Berry, "Quantum Chaology" (The Bakerian Lecture, 1987), Proceedings of the Royal Society of London A 413 (1987) 183-198.

M.V. Berry, "Three quantum obsessions", Nonlinearity 21 (2008) T19–T26

Michael Berry's list of publications (most are available for downloading as PDF files)

M. du Sautoy's account of how Berry came to be involved with the Riemann Hypothesis


partially-successful attempts to produce the required operator/dynamics

This claimed resolution of the Riemann Hypothesis appeared at arXiv.org:

R. Acharya, "Concerning Riemann Hypothesis" (preprint 03/2009)

[abstract:] "We present a quantum mechanical model which establishes the veracity of the Riemann hypothesis that the non-trivial zeros of the Riemann zeta-function lie on the critical line of $\zeta(s)$."

Feedback on this claim is invited (email mwatkins@maths.ex.ac.uk).

Several partially succesful attempts have been made by other researchers to produce the required Hermitian operator, and thereby, via Berry's scheme, prove the Riemann Hypothesis.

H.C. Rosu, "Quantum Hamiltonians and prime numbers", Modern Physics Letters A 18 (2003)

[abstract:] "A short review of Schrödinger hamiltonians for which the spectral problem has been related in the literature to the distribution of the prime numbers is presented here. We notice a possible connection between prime numbers and centrifugal inversions in black holes and suggest that this remarkable link could be directly studied within trapped Bose-Einstein condensates. In addition, when referring to the factorizing operators of Pitkanen and Castro and collaborators, we perform a mathematical extension allowing a more standard supersymmetric approach."

This very welcome, thorough review article discusses and compares the various inter-related work of Bhaduri-Khare-Law, Berry-Keating, Aneva, Castro, et.al., Pitkänen, Khuri, Joffily, Wu-Sprung, Okubo, Mussardo, Boos-Korepin, Crehan and others.

A. Connes, "Formule de trace en geometrie non commutative et hypothese de Riemann", C.R.Sci. Paris, t.323, Serie 1 (Analyse) (1996) 1231-1235.;

(Abstract) "We reduce the Riemann hypothesis for L-functions on a global field k to the validity (not rigorously justified) of a trace formula for the action of the idele class group on the noncommutative space quotient of the adeles of k by the multiplicative group of k."

Berry and Keating refer to this article in their "H = xp and the Riemann zeros", and explain that Connes has devised a Hermitian operator whose eigenvalues are the Riemann zeros on the critical line. This is almost the operator Berry seeks in order to prove the Riemann Hypothesis, but unfortunately the possibility of zeros off the critical line cannot be ruled out in Connes' approach.

His operator is the transfer (Perron-Frobenius) operator of a classical transformation. Such classical operators formally resemble quantum Hamiltonians, but usually have complicated non-discrete spectra and singular eigenfunctions. Connes gets a discrete spectrum by making the operator act on an abstract space where the primes appearing in the Euler product for the Riemann zeta function are built in; the space is constructed from collections of p-adic numbers (adeles) and the associated units (ideles). The proof of the Riemann Hypothesis is thus reduced to the proof of a certain classical trace formula.

A. Deitmar, "A Pólya-Hilbert operator for automorphic L-functions", Indag. Mathem. 12 no.2 (2001) 157-175

[abstract:] "We generalize the first part of A. Connes paper (math/9811068) on the zeroes of the Riemann zeta function from a number field k to any simple algebra M over k. To a given automorphic representation pi of the reductive group M* of invertible elements of M we find a Hilbert space Hpi and an operator Dpi (Pólya-Hilbert operator), which is the infinitesimal generator of a canonical flow such that the spectrum of Dpi coincides with the purely imaginary zeroes of the function L(pi,\rez{2}+z). As a byproduct we get meromorphicity of all automorphic L-functions, not only the cuspidal ones.

S. Okubo, "Lorentz-invariant Hamiltonian and Riemann Hypothesis", J. Phys. A 31 (1998) 1049-1057

"We have given some arguments that a two-dimensional Lorentz-invariant Hamiltonian may be relevant to the Riemann hypothesis concerning zero points of the Riemann zeta function. Some eigenfunction of the Hamiltonian corresponding to infinite-dimensional representation of the Lorentz group have many interesting properties. Especially, a relationship exists between the zero zeta function condition and the absence of trivial representations in the wave function."  

R.K. Bhaduri, Avinash Khare, S.M. Reimann, and E.L. Tomusiak, "The Riemann zeta function and the inverted harmonic oscillator" [outline]

R.K. Bhaduri, Avinash Khare and J. Law, "Phase of the Riemann zeta function and the inverted harmonic oscillator", Physical Review E 52 (1995) 486    [article]       [outline]

"The Argand diagram is used to display some characteristics of the Riemann zeta function...The Argand plots also lead to an analogy with the scattering amplitude and an approximate rule for the location of the zeros. The smooth phase of the zeta function along the line of the zeros is related to the quantum density of states of an inverted oscillator." 

"The loop structure of the zeta function ...with some near-circular shapes, is reminiscent of the Argand plots for the scattering amplitudes of different partial waves in the analysis of resonances, for example, in pion-nucleon scattering." 

J.V. Armitage, "The Riemann Hypothesis and the Hamiltonian of a quantum mechanical system", from Number Theory and Dynamical Systems, eds. M.M. Dodson and J.A.G. Vickers (LMS Lecture Notes, series 134, Cambridge University Press), 153-172. 

"The basic theme of this lecture is an approach to the Riemann Hypothesis in terms of diffusion processes, which has occupied the author's attention for twelve years and which, if correct, has some tantalisingly appealing features culminating in a plausible conjecture that implies the truth of that most celebrated of hypotheses. The connection with diffusion processes suggests that a change of variables (the introduction of imaginary time) might yield a connection with quantum mechanics. That variation offers a possible answer to a conjecture of Berry relating the zeros of the Riemann zeta-function to the Hamiltonian of some quantum mechanical system, which in turn makes precise Hilbert's original suggestion that the zeros are eigenvalues of some operator and the Riemann Hypothesis is true because that operator is Hermitian. We shall offer possible candidates for Hilbert's operator and Berry's Hamiltonian, but we do not claim satisfactorily to have settled those questions, let alone to have proved the Riemann Hypothesis"

Hua Wu and D.W.L. Sprung, "Riemann zeta and a fractal potential", Physical Review E 48 (1993) 2595.

"The nontrivial Riemann zeros are reproduced using a one-dimensional local-potential model. A close look at the potential suggests that it has a fractal structure of dimension d = 1.5."

N.N. Khuri, "Inverse scattering, the coupling constant spectrum, and the Riemann Hypothesis", Math. Phys. Anal. Geom. 5 (2002) 1-63

"We use inverse scattering methods, generalized for a specific class of complex potentials, to construct a one parameter family of complex potentials V(s,r) which have the property that the zero energy s-wave Jost function, as a function of s alone, is identical to Riemann's $\xi$ function whose zeros are the non-trivial zeros of the zeta function. These potentials have an asymptotic expansion in inverse powers of s(s-1) with real coefficients Vn(r) which are explicitly calculated. We show that the validity of the Riemann hypothesis depends essentially on simple integrability properties of the first order coefficient, V1(r). In the case studied in this paper, this coefficient does not satisfy these conditions, but proof of that fact does indicate several possibilities for proceeding further."

K. Chadan and M. Musette, "On an interesting singular potential", C.R. Acad. Sci. Paris 316 II, 1 (1993)

[commentary by H. Rosu:] "Chadan and Musette proposed [a] rather complicated singular potential in a closed interval [0,R] and Dirichlet boundary condition at both ends. They gave arguments that the spectrum in the coupling constant g = 1/4 + t2/4 (t is the imaginary part on the critical line), which is real and discrete, with gn > 1/4 coincides approximately with the nontrivial Riemann zeros when R =exp(-4*pi/3).

We note that this is a so-called Sturmian quantum problem, i.e., a quantization problem in the coupling constant of the potential. A very detailed analysis of this singular hamiltonian and a generalization thereof from the point of view of inverse scattering and s-wave Jost functions has been performed in the important work of Khuri."

R. Acharya, "Realization of the Riemann Hypothesis via coupling constant spectrum" (preprint 03/2008)

[abstract:] "We present a non-relativistic quantum mechanical model, which exhibits the realization of Riemann Conjecture. The technique depends on exposing the $S$-wave Jost function at zero energy and in identifying it with the Riemann $\xi(s)$ function following a seminal paper of N. N. Khuri."

R. Acharya, "Further comments on realization of Riemann Hypothesis via coupling constant spectrum" (preprint 04/2008)

[abstract:] "We invoke Carlson's theorem to justify and to confirm the results previously obtained on the validity of Riemann Hypothesis via the coupling constant spectrum of the zero energy S-wave Jost function a la N. N. Khuri, for the real, repulsive inverse-square potential in non-relativistic quantum mechanics in 3 dimensions."

N. Garcia, "The one-dimensional approachissimo quantum harmonic oscillator: the Hilbert-Pólya Hamiltonian for the primes and the zeros of the Riemann function" (preprint 11/06)

[abstract:] "I have made an ample study of one dimensional quantum oscillators, ranging from logarithmic to exponential potentials. I have found that the eigenvalues of the hamiltonian of the oscillator with the limiting (approachissimo) harmonic potential (~ p(x)2) maps the zeros of the Riemann function height up in the Riemann line. This is the potential created by the field of J(x) that is the Riemann generator of the prime number counting function, p(x), that in turn can be defined by an integral transformation of the Riemann zeta function. This plays the role of the spring strength of the quantum limiting harmonic oscillator. The number theory meaning of this result is that the roots height up of the zeta function are the eigenvalues of a Hamiltonian whose potential is the number of primes squared up to a given x. Therefore this may prove the never published Hilbert-Pólya conjecture. The conjecture is true but does not imply the truth of the Riemann hypothesis. We can have complex conjugated zeros off the Riemman line and map them with another hermitic operator and a general expression is given for that. The zeros off the line affect the fluctuation of the eigenvalues but not their mean values."

Z. Ahmed and S.R. Jain, "A pseudo-unitary ensemble of random matrices, PT-symmetry and the Riemann Hypothesis" (preprint 07/04)

[abstract:] "An ensemble of 2 x 2 pseudo-Hermitian random matrices is constructed that possesses real eigenvalues with level-spacing distribution exactly as for the Gaussian Unitary Ensemble found by Wigner. By a re-interpretation of Connes' spectral interpretation of the zeros of the Riemann zeta function, we propose to enlarge the scope of search of the Hamiltonian connected with the celebrated Riemann Hypothesis by suggesting that the Hamiltonian could also be PT-symmetric (or pseudo-Hermitian)."

Z. Ahmed, "Gaussian-random Ensembles of Pseudo-Hermitian Matrices", Invited Talk Delivered in 2nd International Workshop on 'Pseudo-Hermitian Hanmiltonians in Physics', Prague, June 14–16, 2004

[abstract:] "Attention has been brought to the possibility that statistical fluctuation properties of several complex spectra, or, well-known number sequences may display strong signatures that the Hamiltonian yielding them as eigenvalues is PT-symmetric (Pseudo-Hermitian). We find that the random matrix theory of pseudo-Hermitian Hamiltonians gives rise to new universalities of level-spacing distributions other than those of GOE, GUE and GSE of Wigner and Dyson. We call the new proposals as Gaussian Pseudo-Orthogonal Ensemble and Gaussian Pseudo-Unitary Ensemble. We are also led to speculate that the enigmatic Riemann-zeros would rather correspond to some PT-symmetric (pseudo-Hermitian) Hamiltonian."

M. McGuigan, "Riemann Hypothesis, matrix/gravity correspondence and FZZT brane partition functions" (preprint 08/2007)

[abstract:] "We investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence. The Hilbert-Polya operator in this interpretation is the master matrix of the large N matrix model. Using a related function $\Xi(z)$ we develop an analogy between this function and the Airy function Ai(z) of the Gaussian matrix model. The analogy gives an intuitive physical reason why the zeros lie on a critical line. Using a Fourier transform of the $\Xi(z)$ function we identify a Kontsevich integrand. Generalizing this integrand to $n \times n$ matrices we develop a Kontsevich matrix model which describes n FZZT branes. The Kontsevich model associated with the $\Xi(z)$ function is given by a superposition of Liouville type matrix models that have been used to describe matrix model instantons."

M. McGuigan, "Riemann Hypothesis and master matrix for FZZT brane partition functions" (preprint 05/2008)

[abstract:] "We continue to investigate the physical interpretation of the Riemann zeta function as a FZZT brane partition function associated with a matrix/gravity correspondence begun in arxiv:0708.0645. We derive the master matrix of the $(2,1)$ minimal and $(3,1)$ minimal matrix model. We use it's characteristic polynomial to understand why the zeros of the FZZT partition function, which is the Airy function, lie on the real axis. We also introduce an iterative procedure that can describe the Riemann $\Xi$ function as a deformed minimal model whose deformation parameters are related to a Konsevich integrand. Finally we discuss the relation of our work to other approaches to the Riemann $\Xi$ function including expansion in terms of Meixner-Pollaczek polynomials and Riemann–Hilbert problems."

M. McGuigan, "Riemann hypothesis, modified Morse potential and supersymmetric quantum mechanics" (preprint 02/2020)

[abstract:] "In this paper we discuss various potentials related to the Riemann zeta function and the Riemann Xi function. These potentials are modified versions of Morse potentials and can also be related to modified forms of the radial harmonic oscillator and modified Coulomb potential. We use supersymmetric quantum mechanics to construct their ground state wave functions and the Fourier transform of the ground state to exhibit the Riemann zeros. This allows us to formulate the Riemann hypothesis in terms of the location of the nodes of the ground state wave function in momentum space. We also discuss the relation these potentials to one and two matrix integrals and construct a few orthogonal polynomials associated with the matrix models. We relate the Schr\"odinger equation in momentum space to and finite difference equation in momentum space with an infinite number of terms. We computed the uncertainty relations associated with these potentials and ground states as well as the Shannon Information entropy and compare with the unmodified Morse and harmonic oscillator potentials. Finally we discuss the extension of these methods to other functions defined by a Dirichlet series such as the the Ramanujan zeta function."

S. Matsutani, "On a commutative ring structure in quantum mechanics" (preprint 10/2009)

[abstract:] "In this article, I propose a concept of the $p$-on which is modelled on the multi-photon absorptions in quantum optics. It provides a commutative ring structure in quantum mechanics. Using it, I will give an operator representation of the Riemann $\zeta$ function."

S. Endres and F. Steiner, "The Berry-Keating operator on $L^2(\rz_>,\ud x)$ and on compact quantum graphs with general self-adjoint realizations" (preprint 12/09, submitted to Journal of Physics A)

[abstract:] "The Berry-Keating operator $H_{\mathrm{BK}}:= -\ui\hbar(x\frac{\ud\phantom{x}}{\ud x}+{1/2})$ [M. V. Berry and J. P. Keating, SIAM Rev. 41 (1999) 236] governing the Schr\"odinger dynamics is discussed in the Hilbert space $L^2(\rz_>,\ud x)$ and on compact quantum graphs. It is proved that the spectrum of $H_{\mathrm{BK}}$ defined on $L^2(\rz_>,\ud x)$ is purely continuous and thus this quantization of $H_{\mathrm{BK}}$ cannot yield the hypothetical Hilbert-Polya operator possessing as eigenvalues the nontrivial zeros of the Riemann zeta function. A complete classification of all self-adjoint extensions of $H_{\mathrm{BK}}$ acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of $H_{\mathrm{BK}}$. In addition, an exact trace formula and the Weyl asymptotics of the eigenvalue counting function are derived. Furthermore, we introduce the ``squared'' Berry-Keating operator $H_{\mathrm{BK}}^2:= -x^2\frac{\ud^2\phantom{x}}{\ud x^2}-2x\frac{\ud\phantom{x}}{\ud x}-{1/4}$ which is a special case of the Black-Scholes operator used in financial theory of option pricing. Again, all self-adjoint extensions, the corresponding secular equation, the trace formula and the Weyl asymptotics are derived for $H_{\mathrm{BK}}^2$ on compact quantum graphs. While the spectra of both $H_{\mathrm{BK}}$ and $H_{\mathrm{BK}}^2$ on any compact quantum graph are discrete, their Weyl asymptotics demonstrate that neither $H_{\mathrm{BK}}$ nor $H_{\mathrm{BK}}^2$ can yield as eigenvalues the nontrivial Riemann zeros. Some simple examples are worked out in detail."

K.S. Gupta and E. Harikumar and A.R. de Queiroz, "A Dirac type $xp$-model and the Riemann zeros" (preprint 05/2012)

[abstract:] "We propose a Dirac type modification of the $xp$-model to a $x \slashed{p}$ model on a semi-infinite cylinder. This model is inspired by recent work by Sierra et al. on the $xp$-model on the half-line. Our model realizes the Berry–Keating conjecture on the Riemann zeros. We indicate the connection of our model to that of gapped graphene with a supercritical Coulomb charge, which might provide a physical system for the study of the zeros of the Riemann zeta function."

S. Beltraminelli, D. Merlini and S. Sekatskii, "Riemann Hypothesis: Architecture of a conjecture "along" the lines of Pólya. From trivial zeros and harmonic oscillator to information about non-trivial zeros of the Riemann zeta-function" (preprint 06/2013)

[abstract:] "We propose an architecture of a conjecture concerning the Riemann Hypothesis in the form of an "alternative" to the Pólya strategy: we construct a Hamiltonian $H_{Polya}$ whose spectrum coincides exactly with that of the harmonic oscillator Hamiltonian $H_{osc}$ if and only if the Riemann Hypothesis holds true. In other words, it can be said that we formulate the Riemann Hypothesis by means of a non-commutative structure on the real axis, viz., that of the harmonic oscillator, by an equation of the type $H_{Polya}(H_{osc}) = H_{osc}$: the harmonic oscillator operator, if viewed as an argument of $H_{Polya}, reproduces itself."

G. Menezes and N.F. Svaiter, "Quantum field theories and prime numbers spectrum" (preprint 11/2012)

[abstract:] "The Riemann hypothesis states that all nontrivial zeros of the zeta function lie on the critical line $\Re(s)=1/2$. Hilbert and Pólya suggested a possible approach to prove it, based on spectral theory. Within this context, some authors formulated the question: is there a quantum mechanical system related to the sequence of prime numbers? In this Letter we assume that there is a class of hypothetical physical systems described by self-adjoint operators with countable infinite number of degrees of freedom with spectra given by the sequence of primes numbers. We prove a no-go theorem. We show that the generating functional of connected Schwinger functions of such theories cannot be constructed."

G. Menezes, B.F. Svaiter and N.F. Svaiter, "Zeta regularized products, Riemann zeta zeros and prime number spectra" (preprint 03/2013)

[abstract:] "The Riemann hypothesis states that all nontrivial zeros of the zeta function lie in the critical line $\Re(s)=1/2$. Hilbert and P\'olya suggested that one possible way to prove the Riemann hypothesis is to interpret the nontrivial zeros in the light of spectral theory. Following this approach, we associate such a numerical sequence with the discrete spectrum of a linear differential operator. We discuss a necessary condition that such a sequence of numbers should obey in order to be associated with the spectrum of a linear differential operator of a system with countably infinite number of degrees of freedom. The sequence of nontrivial zeros is zeta regularizable. Then, functional integrals associated with hypothetical systems described by self-adjoint operators whose spectra is given by the sequence of the nontrivial zeros of the Riemann zeta function could be constructed. In addition, we demonstrate that if one considers the same situation with primes numbers, the associated functional integral cannot be constructed, due to the fact that the sequence of prime numbers is not zeta regularizable."

C.M. Bender, D.C. Brody and M.P. Müller, "Hamiltonian for the zeros of the Riemann zeta function", Phys. Rev. Lett. 118 (2017) 130201

[abstract:] "A Hamiltonian operator $\hat{H}$ is constructed with the property that if the eigenfunctions obey a suitable boundary condition, then the associated eigenvalues correspond to the nontrivial zeros of the Riemann zeta function. The classical limit of $\hat{H}$ is $2xp$, which is consistent with the Berry–Keating conjecture. While $\hat{H}$ is not Hermitian in the conventional sense, $i\hat{H}$ is $\mathcal{PT}$ symmetric with a broken $\mathcal{PT}$ symmetry, thus allowing for the possibility that all eigenvalues of $\hat{H}$ are real. A heuristic analysis is presented for the construction of the metric operator to define an inner-product space, on which the Hamiltonian is Hermitian. If the analysis presented here can be made rigorous to show that $\hat{H}$ is manifestly self-adjoint, then this implies that the Riemann hypothesis holds true."

C.M. Bender and D.C. Brody, "Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian" (preprint 11/2017)

"The differential equation eigenvalue problem associated with a recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of the Riemann zeta function, is analyzed using the Fourier and the WKB analysis. The Fourier analysis leads to an open problem concerning the formulation of the eigenvalue problem in the momentum space. The WKB analysis gives an exact asymptotic behavior of the eigenfunction."

N. Wolchover, "Physicists attack math's $1,000,000 question", Quanta (4 April 2017) [somewhat misleading popular science article about the publication by Bender, et al.]

J.V. Bellissard, "Comment on "Hamiltonian for the zeros of the Riemann zeta function"" (preprint, 04/2017)

[abstract:] "This comment about the article "Hamiltonian for the Zeros of the Riemann Zeta Function", by C. M. Bender, D. C. Brody, and M. P. Müller, published recently in Phys. Rev. Lett. (Phys. Rev. Lett., 118, 130201, (2017)) gives arguments showing that the strategy proposed by the authors to prove the Riemann Hypothesis, does not actually work."

M. Bishop, E. Aiken and D. Singleton, "Modified commutation relationships from the Berry–Keating program" (preprint 10/2018)

[abstract:] "Current approaches to quantum gravity suggest there should be a modification of the standard quantum mechanical commutator, $[\hat{\mathbf{x}},\hat{\mathbf{p}}] = i\hbar$. Typical modifications are phenomenological and designed to result in a minimal length scale. As a motivating principle for the modification of the position and momentum commutator, we assume the validity of a version of the Bender–Brody–Müller variant of the Berry–Keating approach to the Riemann hypothesis. We arrive at a family of modified position and momentum operators, and their associated modified commutator, which lead to a minimal length scale. Additionally, this larger family generalizes the Bender–Brody–Müller approach to the Riemann hypothesis."

F.I. Moxley III, "Solving the Riemann Hypothesis with Green's function and a Gelfand triplet" (preprint 06/2018)

[abstract:] "The Hamiltonian of a quantum mechanical system has an aliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender–Brody–Müller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a Hermitian Hamiltonian that maps to the nontrivial zeros of the analytic continuation of the Riemann zeta function, and provide an analytical expression for the eigenvalues of the results using Green's functions. A Gelfand triplet is then used to ensure that the eigenvalues are well defined. The holomorphicity of the resulting eigenvalues is demonstrated, and it is shown that that the expectation value of the Hamiltonian operator is also zero such that the nontrivial zeros of the Riemann zeta function are not observable. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function exists at $\sigma = 1/2$, and a general solution is obtained by performing an invariant similarity transformation."

F.I. Moxley III, "Decidability of the Riemann Hypothesis" (preprint 09/2018)

[abstract:] "The Hamiltonian of a quantum mechanical system has an affiliated spectrum, and in order for this spectrum to be observable, the Hamiltonian should be Hermitian. Such quantum Riemann zeta function analogies have led to the Bender–Brody–Müller (BBM) conjecture, which involves a nonHermitian Hamiltonian whose eigenvalues are the nontrivial zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a Hermitian Hamiltonian using a similarity transformation, and provide an analytical expression for the eigenvalues of the results using Green's functions. A Gelfand triplet is then used to ensure that the eigenvalues are well defined. The holomorphicity of the resulting eigenvalues is demonstrated, and it is shown that that the expectation value of the Hamiltonian operator is also zero such that the nontrivial zeros of the Riemann zeta function are not observable, i.e., the Riemann Hypothesis is not decidable. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function exists at $\sigma = 1/2$, and a general solution is obtained by performing an invariant similarity transformation"

F.I. Moxley III, "A Schrödinger equation for solving the Bender–Brody–Muller Conjecture" (AIP Conference Proceedings, 2017)

[abstract:] "The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender–Brody–Muller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and discuss the eigenvalues of the results. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, the Hilbert–Pólya conjecture is discussed, and it is heuristically shown that the real part of every nontrivial zero of the Riemann zeta function converges at $\sigma = 1/2$."

M.C. Nucci, "Spectral realization of the Riemann zeros by quantizing $H = w(x)(p + \ell_2_p/p)$: the Lie–Noether symmetry approach", Journal of Physics: Conference Series 482, conference 1

[abstract:] "If $t_n$ are the heights of the Riemann zeros $1/2 + it_n$, an old idea, attributed to Hilbert and Pólya, stated that the Riemann hypothesis would be proved if the $t_n$ could be shown to be eigenvalues of a self-adjoint operator. In 1986 Berry conjectured that $t_n$ could instead be the eigenvalues of a deterministic quantum system with a chaotic classical counterpart and in 1999 Berry and Keating proposed the Hamiltonian $H = xp$, with $x$ and $p$ the position and momentum of a one-dimensional particle, respectively. This was proven not to be the correct Hamiltonian since it yields a continuum spectrum and therefore a more general Hamiltonian $H = w(x)(p + \ell_2p/p)$ was proposed and different expressions of the function $w(x)$ were considered although none of them yielding exactly $t_n$. We show that the quantization by means of Lie and Noether symmetries of the Lagrangian equation corresponding to the Hamiltonian $H$ yields straightforwardly the Schröodinger equation and clearly explains why either the continuum or the discrete spectrum is obtained. Therefore we infer that suitable Lie and Noether symmetries of the classical Lagrangian corresponding to $H$ should be searched in order to alleviate one of Berry's quantum obsessions."

A. Chattopadhyay, P. Dutta, S. Dutta and D. Ghoshal, "Matrix model for Riemann zeta via its local factors" (preprint 07/2018)

[abstract:] "We propose the construction of an ensemble of unitary random matrices (UMM) for the Riemann zeta function. Our approach to this problem is 'piecemeal', in the sense that we consider each factor in the Euler product representation of the zeta function to first construct a UMM for each prime $p$. We are able to use its phase space description to write the partition function as the trace of an operator that acts on a subspace of square-integrable functions on the $p$-adic line. This suggests a Berry–Keating type Hamiltonian. We combine the data from all primes to propose a Hamiltonian and a matrix model for the Riemann zeta function."

D. Lebiedz ,"Holomorphic Hamiltonian $\xi$-flow and Riemann zeros"

[abstract:] "With a view on the formal analogy between the Riemann–von Mangoldt explicit formula and semiclassical quantum mechanics in terms of the Gutzwiller trace formula we construct a complex-valued Hamiltonian $H(q,p)=\xi(q)p$ from the holomorphic flow $q˙=\xi(q)$ and its variational differential equation. The Hamiltonian phase portrait $q(p)$ is a Riemann surface equivalent to reparameterized $\xi$-Newton flow solutions in complex-time, its flow map differential is determined by all Riemann zeros and reminiscent of a `spectral sum' in trace formulas. Canonical quantization for particle quantum mechanics on a circle leads to a Dirac-type momentum operator with discrete spectrum given by classical closed orbit periods determined by derivatives $\xi'(\rho_n)$ at Riemann zeros."


other related material

There are a number of other approaches:

R. He, M.-Z. Ai, J.-M. Cui, Y.-F. Huang, Y.-J. Han, C.-F. Li and G.-C. Guo, "Finding the Riemann zeros by periodically driving a single trapped ion" (preprint 03/2019)

[abstract:] "The Riemann hypothesis implies the most profound secret of the prime numbers. It is still an open problem despite various attempts have been made by numerous mathematicians. One of the most fantastic approaches to treat this problem is to connect this hypothesis with the spectrum of a physical Hamiltonian. However, designing and performing a suitable Hamiltonian corresponding to this conjecture is always a primary challenge. Here we report the first experiment to find the non-trivial zeros of the Riemann function and Pólya's function using the novel approach proposed by Floquet method. In this approach, the zeros of the functions instead are characterized by the occurance of the crossings of the quasienergies when the dynamics of the system is frozen. With the properly designed periodically driving functions, we can experimentally obtain the first non-trivial zero of the Riemann function and the first two non-trivial zeros of Pólya's function which are in excellent agreement with their exact values. Our work provides a new insight for the Pólya--Hilbert conjecture in quantum systems."

R.S. Mackay, "Towards a spectral proof of Riemann's hypothesis" (preprint 08/2017)

[abstract:] "The paper presents evidence that Riemann's $\xi$ function evaluated at $2\sqrt(E)$ could be the characteristic function $P(E)$ for the magnetic Laplacian minus $85/16$ on a surface of curvature $-1$ with magnetic field $9/4$, a cusp of width $1$, a Dirichlet condition at a point, and other conditions not yet determined."

R. Sasaki, "Symmetric Morse potential is exactly solvable" (preprint, 11/2016)

[abstract:] "Morse potential $V_M(x)= g^2\exp (2x)-g(2h+1)\exp(x)$ is defined on the full line, $-\infty<x<\infty$ and it defines an exactly solvable 1-d quantum mechanical system with finitely many discrete eigenstates. By taking its right half $0\le x<\infty$ and glueing it with the left half of its mirror image $V_M(-x)$, $-\infty<x<0$, the symmetric Morse potential $V(x)= g^2\exp (2|x|)-g(2h+1)\exp(|x|)$ is obtained. The quantum mechanical system of this piecewise analytic potential has infinitely many discrete eigenstates with the corresponding eigenfunctions given by the Whittaker W function. The eigenvalues are the square of the zeros of the Whittaker function $W_{k,\nu}(x)$ and its linear combination with $W'_{k,\nu}(x)$ as a function of $\nu$ with fixed $k$ and $x$. This quantum mechanical system seems to offer an interesting example for discussing the Hilbert--P\'olya conjecture on the pure imaginary zeros of Riemann zeta function on $\Re(s)=\frac{1}{2}$."

J. Berra-Montiel and A. Molgado, "Polymeric quantum mechanics and the zeros of the Riemann zeta function" (preprint, 10/2016)

[abstract:] "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 Riemann–von Mangoldt formula, and introduces a correction depending on the energy and the scale parameter, which resembles the fluctuation behavior of the Riemann zeros."

J. Kuipers, Q. Hummel and K. Richter, "Quantum graphs whose spectra mimic the zeros of the Riemann zeta function" (preprint 07/2013)

[abstract:] "One of the most famous problems in mathematics is the Riemann hypothesis: that the non-trivial zeros of the Riemann zeta function lie on a line in the complex plane. One way to prove the hypothesis would be to identify the zeros as eigenvalues of a Hermitian operator, many of whose properties can be derived through the analogy to quantum chaos. Using this, we construct a set of quantum graphs that have the same oscillating part of the density of states as the Riemann zeros, offering an explanation of the overall minus sign. The smooth part is completely different, and hence also the spectrum, but the graphs pick out the low-lying zeros."

B. Barrau, "On Hilbert–Pólya conjecture: Hermitian operator naturally associated to L-functions" (preprint 05/2011)

[abstract:] "Using as starting point a classical integral representation of a L-function we define a family of two variables extended functions which are eigenfunctions of a Hermitian operator (having imaginary part of zeros as eigenvalues). This Hermitian operator can take also other forms, more symetric. In the case of particular L-functions, like Zeta function or Dirichlet L-functions, the eigenfunctions defined for this operator have symmetry properties. Moreover, for s zero of Zeta function (or Dirichlet L-function), the associated eigenfunction has a specific property (a part of eigenfunction is cancelled). Finding such an eigenfunction, square integrable due to this "cancellation effect", would lead to Riemann Hypothesis using Hilbert–Pólya idea."

M. Srednicki, "Nonclasssical Degrees of Freedom in the Riemann Hamiltonian" (preprint 05/2011)

[abstract:] "The Hilbert–Pólya conjecture states that the imaginary parts of the zeros of the Riemann zeta function are eigenvalues of a quantum hamiltonian. If so, conjectures by Katz and Sarnak put this hamiltonian in Altland and Zirnbauer's universality class C. This implies that the system must have a nonclassical two-valued degree of freedom. In such a system, the dominant primitive periodic orbits contribute to the density of states with a phase factor of –1, which partially resolves a previously mysterious sign problem for oscillatory contributions to the density of the Riemann zeros."

Yang-Hui He, V. Jejjala, D. Minic, "On the physics of the Riemann zeros" (Quantum Theory and Symmetries 6 conference proceedings)

[abstract:] "We discuss a formal derivation of an integral expression for the Li coefficients associated with the Riemann xi-function which, in particular, indicates that their positivity criterion is obeyed, whereby entailing the criticality of the non-trivial zeros. We conjecture the validity of this and related expressions without the need for the Riemann Hypothesis and discuss a physical interpretation of this result within the Hilbert-Polya approach. In this context we also outline a relation between string theory and the Riemann Hypothesis."

G. Regniers, "Wigner quantization of some one-dimensional Hamiltonians" (preprint 11/2010)

[abstract:] "Recently, several papers have been dedicated to the Wigner quantization of different Hamiltonians. In these examples, many interesting mathematical and physical properties have been shown. Among those we have the ubiquitous relation with Lie superalgebras and their representations. In this paper, we study two one-dimensional Hamiltonians for which the Wigner quantization is related with the orthosymplectic Lie superalgebra osp(1|2). One of them, the Hamiltonian H = xp, is popular due to its connection with the Riemann zeros, discovered by Berry and Keating on the one hand and Connes on the other. The Hamiltonian of the free particle, H_f = p^2/2, is the second Hamiltonian we will examine. Wigner quantization introduces an extra representation parameter for both of these Hamiltonians. Canonical quantization is recovered by restricting to a specific representation of the Lie superalgebra osp(1|2)."

P. Ranjan Giri and R.K. Bhaduri, "Physical interpretation for Riemann zeros from black hole physics" (preprint 05/2009)

[abstract:] "According to a conjecture attributed to Poyla and Hilbert, there is a self-adjoint operator whose eigenvalues are the the nontrivial zeros of the Riemann zeta function. We show that the near-horizon dynamics of a massive scalar field in the Schwarzschild black hole spacetime, under a reasonable boundary condition, gives rise to energy eigenvalues that coincide with the Riemann zeros. In achieving this result, we exploit the Bekenstein conjecture of black hole area quantization, and argue that it is responsible for the breaking of the continuous scale symmetry of the near horizon dynamics into a discrete one."

P. Betzios, N. Gaddam and O. Papadoulaki, "Black holes, quantum chaos, and the Riemann hypothesis" (preprint 04/2020)

[abstract:] "Quantum gravity is expected to gauge all global symmetries of effective theories, in the ultraviolet. Inspired by this expectation, we explore the consequences of gauging CPT as a quantum boundary condition in phase space. We find that it provides for a natural semiclassical regularisation and discretisation of the continuous spectrum of a quantum Hamiltonian related to the Dilation operator. We observe that the said spectrum is in correspondence with the zeros of the Riemann zeta and Dirichlet beta functions. Following ideas of Berry and Keating, this may help the pursuit of the Riemann hypothesis. It strengthens the proposal that this quantum Hamiltonian captures the dynamics of the scattering matrix on a Schwarzschild black hole background, given the rich chaotic spectrum upon discretisation. It also explains why the spectrum appears to be erratic despite the unitarity of the scattering matrix."

G. Sierra, "The Riemann zeros and the cyclic Renormalization Group" (preprint 10/2005)

[abstract:] "We propose a consistent quantization of the Berry-Keating Hamiltonian xp, which is currently discussed in connection with the non trivial zeros of the Riemann zeta function. The smooth part of the Riemann counting formula of the zeros is reproduced exactly. The zeros appear, not as eigenstates, but as missing states in the spectrum, in agreement with Connes adelic approach to the Riemann hypothesis. The model is exactly solvable and renormalizable, with a cyclic Renormalization Group. These results are obtained by mapping the Berry-Keating model into the Russian doll model of superconductivity. Finally, we propose a generalization of these models in an attempt to explain the oscillatory part of the Riemann's formula."

G. Sierra, "H = xp with interaction and the Riemann zeros", Nucl. Phys. B 776 (3) (2007) 327–364

[abstract:] "Starting from a quantized version of the classical Hamiltonian $H=xp$, we add a non-local interaction which depends on two potentials. The model is solved exactly in terms of a Jost like function which is analytic in the complex upper half plane. This function vanishes, either on the real axis, corresponding to bound states, or below it, corresponding to resonances. We find potentials for which the resonances converge asymptotically toward the average position of the Riemann zeros. These potentials realize, at the quantum level, the semiclassical regularization of $H=xp$ proposed by Berry and Keating. Furthermore, a linear superposition of them, obtained by the action of integer dilations, yields a Jost function whose real part vanishes at the Riemann zeros and whose imaginary part resembles the one of the zeta function. Our results suggest the existence of a quantum mechanical model where the Riemann zeros would make a point like spectrum embedded in the continuum. The associated spectral interpretation would resolve the emission/absorption debate between Berry–Keating and Connes. Finally, we indicate how our results can be extended to the Dirichlet L-functions constructed with real characters."

G. Sierra, "On the quantum reconstruction of the Riemann zeros" (preprint 11/2007)

[abstract:] "We discuss a possible spectral realization of the Riemann zeros based on the Hamiltonian $H = xp$ perturbed by a term that depends on two potentials, which are related to the Berry-Keating semiclassical constraints. We find perturbatively the potentials whose Jost function is given by the zeta function $\zeta(\sigma - i t)$ for $\sigma > 1$. For $\sigma = 1/2$ we find the potentials that yield the smooth approximation to the zeros. We show that the existence of potentials realizing the zeta function at $\sigma = 1/2$, as a Jost function, would imply that the Riemann zeros are point like spectrum embedded in the continuum, resolving in that way the emission/spectral interpretation."

G. Sierra, "A quantum mechanical model of the Riemann zeros" (preprint 12/2007)

[abstract:] "In 1999 Berry and Keating showed that a regularization of the 1D classical Hamiltonian H = xp gives semiclassically the smooth counting function of the Riemann zeros. In this paper we first generalize this result by considering a phase space delimited by two boundary functions in position and momenta, which induce a fluctuation term in the counting of energy levels. We next quantize the xp Hamiltonian, adding an interaction term that depends on two wave functions associated to the classical boundaries in phase space. The general model is solved exactly, obtaining a continuum spectrum with discrete bound states embbeded in it. We find the boundary wave functions, associated to the Berry-Keating regularization, for which the average Riemann zeros become resonances. A spectral realization of the Riemann zeros is achieved exploiting the symmetry of the model under the exchange of position and momenta which is related to the duality symmetry of the zeta function. The boundary wave functions, giving rise to the Riemann zeros, are found using the Riemann-Siegel formula of the zeta function. Other Dirichlet L-functions are shown to find a natural realization in the model."

G. Sierra, P.K. Townsend, "Landau levels and Riemann zeros" (preprint 05/2008)

[abstract:] "The number $N(E)$ of complex zeros of the Riemann zeta function with positive imaginary part less than $E$ is the sum of a 'smooth' function $\bar N(E)$ and a 'fluctuation'. Berry and Keating have shown that the asymptotic expansion of $\bar N(E)$ counts states of positive energy less than $E$ in a 'regularized' semi-classical model with classical Hamiltonian $H=xp$. For a different regularization, Connes has shown that it counts states 'missing' from a continuum. Here we show how the 'absorption spectrum' model of Connes emerges as the lowest Landau level limit of a specific quantum mechanical model for a charged particle on a planar surface in an electric potential and uniform magnetic field. We suggest a role for the higher Landau levels in the fluctuation part of $N(E)$."

G. Sierra and J. Rodriguez-Lagunam, "The $H = xp$ model revisited and the Riemann zeros" (preprint 02/2011)

[abstract:] "Berry and Keating conjectured that the classical Hamiltonian $H = xp$ is related to the Riemann zeros. A regularization of this model yields semiclassical energies that behave, in average, as the non trivial zeros of the Riemann zeta function. However, the classical trajectories are not closed, rendering the model incomplete. In this paper, we show that the Hamiltonian $H = x(p + l_p^2/p)$ contains closed periodic orbits, and that its spectrum coincides with the average Riemann zeros. This result is generalized to Dirichlet L-functions using different self-adjoint extensions of H. We discuss the relation of our work to Pólya's fake zeta function and suggest an experimental realization in terms of the Landau model."

G. Sierra, "General covariant $xp$ models and the Riemann zeros" (preprint 10/2011)

[abstract:] "We study a general class of models whose classical Hamiltonians are given by $H = U(x) p + V(x)/p$, where $x$ and $p$ are the position and momentum of a particle moving in one dimension, and $U$ and $V$ are positive functions. This class includes the Hamiltonians $H_I = x(p + 1/p)$ and $H_{II}=(x + 1/x)(p + 1/p)$, which have been recently discussed in connection with the non trivial zeros of the Riemann zeta function. We show that all these models are covariant under general coordinate transformations. This remarkable property becomes explicit in the Lagrangian formulation which describes a relativistic particle moving in a $1+1$ dimensional spacetime whose metric is constructed from the functions $U$ and $V$. General covariance is maintained by quantization and we find that the spectra are closely related to the geometry of the associated spacetimes. In particular, the Hamiltonian $H_I$ corresponds to a flat spacetime, whereas its spectrum approaches the Riemann zeros in average. The latter property also holds for the model $H_{II}$, whose underlying spacetime is asymptotically flat. These results suggest the existence of a Hamiltonian whose underlying spacetime encodes the prime numbers, and whose spectrum provides the Riemann zeros."

G. Sierra, "The Riemann zeros as energy levels of a Dirac fermion in a potential built from the prime numbers in Rindler spacetime" (preprint 04/2014)

[abstract:] "We construct a Hamiltonian $H$ whose discrete spectrum contains, in a certain limit, the Riemann zeros. $H$ is derived from the action of a massless Dirac fermion living in a domain of Rindler spacetime, in 1+1 dimensions, that has a boundary given by the world line of a uniformly accelerated observer.

The action contains a sum of delta function potentials that can be viewed as partially reflecting moving mirrors. An appropriate choice of the accelerations of the mirrors, provide primitive periodic orbits associated to the prime numbers p, whose periods, measured by the observer's clock, are $\log p$. Acting on the chiral components of the fermion, $H$ becomes the Berry–Keating Hamiltonian $(xp + px)/2$, where $x$ is identified with the Rindler spatial coordinate and $p$ with the conjugate momentum.

The delta function potentials give the matching conditions of the fermion wave functions on both sides of the mirrors. There is also a phase shift for the reflection of the fermions at the boundary where the observer sits. The eigenvalue problem is solved by transfer matrix methods in the limit where the reflection amplitudes become infinitesimally small. We find that for generic values of the phase shift the spectrum is a continuum, where the Riemann zeros are missing, as in the adelic Connes model. However, for some values of phase shift, related to the phase of the zeta function, the Riemann zeros appear as discrete eigenvalues immersed in the continuum. We generalize this result to the zeros of Dirichlet $L$-functions, associated to primitive characters, that are encoded in the reflection coefficients of the mirrors. Finally, we show that the Hamiltonian associated to the Riemann zeros belongs to class AIII, or chiral GUE, of Random Matrix Theory."

C.E. Creffield and G. Sierra, "Finding zeros of the Riemann zeta function by periodic driving of cold atoms" (preprint 11/2014)

[abstract:] "The Riemann hypothesis, which states that the non-trivial zeros of the Riemann zeta function all lie on a certain line in the complex plane, is one of the most important unresolved problems in mathematics. Inspired by the Pólya–Hilbert conjecture, we propose a new approach to finding a physical system to study the Riemann zeros, which in contrast to previous examples, is based on applying a time-periodic driving field. This driving allows us to mould the quasienergies of the system (the analogue of the eigenenergies in the absence of driving), so that they are directly governed by the zeta function. We further show by numerical simulations that this allows the Riemann zeros to be measured in currently accessible cold atom experiments."

G. Sierra, "The Riemann zeros as spectrum and the Riemann hypothesis" (preprint 01/2016)

"We review a series of works whose aim is to provide a spectral realization of the Riemann zeros and that culminate in a physicist's proof of the Riemann hypothesis. These results are obtained analyzing the spectrum of the Hamiltonian of a massless Dirac fermion in a region of Rindler spacetime that contains moving mirrors whose accelerations are related to the prime numbers. We show that a zero on the critical line becomes an eigenvalue of the Hamiltonian in the limit where the mirrors become transparent, and the self-adjoint extension of the Hamiltonian is adjusted accordingly with the phase of the zeta function. We have also considered the spectral realization of zeros off the critical line using a non self-adjoint operator, but its properties imply that those zeros do not exist. In the derivation of these results we made several assumptions that need to be established more rigorously."

J. Andrade, "Hilbert–Pólya conjecture, zeta-functions and bosonic quantum field theories" (preprint 05/2013)

[abstract:] "The original Hilbert and Pólya conjecture is the assertion that the non-trivial zeros of the Riemann zeta function can be the spectrum of a self-adjoint operator. So far no such operator was found. However the suggestion of Hilbert and Pólya, in the context of spectral theory, can be extended to approach other problems and so it is natural to ask if there is a quantum mechanical system related to other sequences of numbers which are originated and motivated by number theory.

In this paper we show that the functional integrals associated with a hypothetical class of physical systems described by self-adjoint operators associated with bosonic fields whose spectra is given by three different sequence of numbers cannot be constructed. The common feature of the sequence of numbers considered here, which causes the impossibility of zeta regularization, is that the various Dirichlet series attached to such sequences – such as those which are sums over "primes" of $(\mathrm{norm}\ P)^{-s}$ have a natural boundary, i.e., they cannot be continued beyond the line $\Re(s)=0$. The main argument is that once the regularized determinant of a Laplacian is meromorphic in $s$, it follows that the series considered above cannot be a regularized determinant. In other words we show that the generating functional of connected Schwinger functions of the associated quantum field theories cannot be constructed."

G. Regniers and J. Van der Jeugt, "The Hamiltonian $H=xp$ and classification of $osp(1|2)$ representations" (Contribution for the Workshop Lie Theory and Its Applications in Physics VIII, Varna, 2009)

[abstract:] "The quantization of the simple one-dimensional Hamiltonian $H=xp$ is of interest for its mathematical properties rather than for its physical relevance. In fact, the Berry-Keating conjecture speculates that a proper quantization of $H=xp$ could yield a relation with the Riemann hypothesis. Motivated by this, we study the so-called Wigner quantization of $H=xp$, which relates the problem to representations of the Lie superalgebra $osp(1|2)$. In order to know how the relevant operators act in representation spaces of $osp(1|2)$, we study all unitary, irreducible star representations of this Lie superalgebra. Such a classification has already been made by J.W.B. Hughes, but we reexamine this classification using elementary arguments."

M. Srednicki, "The Berry–Keating Hamiltonian and the Local Riemann Hypothesis" (preprint 04/2011)

[abstract:] "The local Riemann hypothesis states that the zeros of the Mellin transform of a harmonic-oscillator eigenfunction (on a real or $p$-adic configuration space) have real part 1/2. For the real case, we show that the imaginary parts of these zeros are the eigenvalues of the Berry–Keating hamiltonian $H=(xp+px)/2$ projected onto the subspace of oscillator eigenfunctions of lower level. This gives a spectral proof of the local Riemann hypothesis for the reals, in the spirit of the Hilbert–Pólya conjecture. The $p$-adic case is also discussed."

D. Bump, Kwok-Kwong Choi, P. Kurlberg, and J. Vaaler, "A Local Riemann Hypothesis, I", Mathematische Zeitschrift 233 (1) (2000), 1-18. (A subscription to Mathematische Zeitschrift is required if you wish to download this.)

"[This paper describes] how local Tate integrals formed with eigenfunctions of the quantum mechanical harmonic oscillator, and its p-adic analogs, have their zeros on the line Re(s) = 1/2. This...incorporates new material on the harmonic oscillator in n dimensions, Mellin transforms of the Laguerre functions, and a reciprocity law for their values at negative integers."

B. Aneva, "Symmetry in phase space of a chaotic system", from AIP conference proceedings Disordered and Complex Systems 553 (Springer, 2001) 173–178

[abstract:] "Finite symmetry in phase space is used for a geometrical interpretation of chaos quantization conditions, which relate the eigenvalues of a Hamiltonian operator with the non-trivial zeros of the Riemann zeta function."

B. Aneva, "Symmetry of the Riemann operator", Physics Letters B 450 (1999) 388-396.

[abstract:] "Chaos quantization conditions, which relate the eigenvalues of a Hermitian operator (the Riemann operator) with the non-trivial zeros of the Riemann zeta function are considered, and their geometrical interpretation is discussed"

D. Schumayer, B.P. van Zyl and D.A.W. Hutchinson, "Quantum mechanical potentials related to the prime numbers and Riemann zeros", Phys. Rev. E 78 (2008) 056215

[abstract:] "Prime numbers are the building blocks of our arithmetic; however, their distribution still poses fundamental questions. Riemann showed that the distribution of primes could be given explicitly if one knew the distribution of the nontrivial zeros of the Riemann $\zeta(s)$ function. According to the Hilbert–Pólya conjecture, there exists a Hermitian operator of which the eigenvalues coincide with the real parts of the nontrivial zeros of $\zeta(s)$. This idea has encouraged physicists to examine the properties of such possible operators, and they have found interesting connections between the distribution of zeros and the distribution of energy eigenvalues of quantum systems. We apply the Marchenko approach to construct potentials with energy eigenvalues equal to the prime numbers and to the zeros of the $\zeta(s)$ function. We demonstrate the multifractal nature of these potentials by measuring the Rényi dimension of their graphs. Our results offer hope for further analytical progress."

J. Sakhr, R.K. Bhaduri and B.P. van Zyl, "Zeta function zeros, powers of primes, and quantum chaos" Phys. Rev. E 68 (2003) 026105

[abstract:] "We present a numerical study of Riemann's formula for the oscillating part of the density of the primes and their powers. The formula is comprised of an infinite series of oscillatory terms, one for each zero of the zeta function on the critical line and was derived by Riemann in his paper on primes assuming the Riemann hypothesis. We show that high resolution spectral lines can be generated by the truncated series at all powers of primes and demonstrate explicitly that the relative line intensitites are correct. We then derive a Gaussian sum rule for Riemann's formula. This is used to analyze the numerical convergence of the truncated series The connections to quantum chaos and semiclassical physics are discussed."

B.P. van Zyl and D.A.W. Hutchinson, "Riemann zeros, prime numbers, and fractal potentials", Phys. Rev. E 67 (2003)

[abstract:] "Using two distinct inversion techniques, the local one-dimensional potentials for the Riemann zeros and prime number sequence are reconstructed. We establish that both inversion techniques, when applied to the same set of levels, lead to the same fractal potential. This provides numerical evidence that the potential obtained by inversion of a set of energy levels is unique in one dimension. We also investigate the fractal properties of the reconstructed potentials and estimate the fractal dimensions to be D = 1.5 for the Riemann zeros and D = 1.8 for the prime numbers. This result is somewhat surprising since the nearest-neighbor spacings of the Riemann zeros are known to be chaotically distributed, whereas the primes obey almost Poissonlike statistics. Our findings show that the fractal dimension is dependent on both level statistics and spectral rigidity, \Delta3, of the energy levels."

P. Crehan, "Chaotic spectra of classically integrable systems", Journal of Physics A 28 6389-6394

"We prove that any spectral sequence obeying a certain growth law is the quantum spectrum of an equivalence class of classically integrable non-linear oscillators. This implies that exceptions to the Berry-Tabor rule for the distribution of quantum energy gaps of clasically integrable systems, are far more numerous than previously believed. In particular we show that for each finite dimension k, there are an infinite number of classically integrable k-dimensional non-linear oscillators whose quantum spectrum reproduces the imaginary part of zeros on the critical line of the Riemann zeta function."

J.J. Garcia Moreta, "Approximate solutions of the Urysohn integral equation with a nonlinearity of the form $K(u,x,\phi(x)) = R(u,x)e^{iu\phi(x)}$ and its connection with the Riemann Hypothesis" (preprint, 2007)

[abstract:] "In this paper we study th eapproximate solutions to solve the Urysohn integral equation of first and second kind with an exponential nonlinearity. This study is motivated due to reinterpretation of the Chebyshev function in number theory as the trace of certain Hamiltonian operator, which in the end yields to the fact that the potential $V(x)$ of a Hamiltonian whose energies are just the non-trivial zeros of the Riemann zeta function is the solution of a certain Urysohn equation of first kind. Also we discuss the relationship between these type of integrals and linear PDE and ODE with certain initial value or boundary conditions and propose a new Hilbert-Pólya operator (Hamiltonian) as a possible candidate to satisfy Riemann Hypothesis"

There has been some controversy involving the removal of this author's earlier preprints from arXiv.org. They were "withdrawn by arXiv administrators because of fraudulently claimed institutional affiliation and status". Here, Moreta claims that this was due to a misunderstanding.

J.-F. Burnol, "On some bound and scattering states associated with the cosine kernel" (preprint 01/2008)

[abstract:] "It is explained how to provide self-adjoint operators having scattering states forming a multiplicity one continuum and bound states whose corresponding eigenvalues have an asymptotic density equivalent to the one of the zeros of the Riemann zeta function. It is shown how this can be put into an integro-differential form of a type recently considered by Sierra."

J.-F. Burnol, "A lower bound in an approximation problem involving the zeros of the Riemann zeta function", Advances in Mathematics 170 (2002) 56-70

[Abstract:] "We slightly improve the lower bound of Baez-Duarte, Balazard, Landreau and Saias in the Nyman-Beurling formulation of the Riemann Hypothesis as an approximation problem. We construct Hilbert space vectors which could prove useful in the context of the the so called 'Hilbert-Pólya idea'."

S. Joffily, "A model for the quantum vacuum", Nucl. Phys. A 790 (1-4) (2007) 631c–634c

[abstract:] "Following our recent works [S. Joffily, Jost function, prime numbers and Riemann zeta function, Contribution to Roberto Salmeron Festschrift, eds. by R. Aldrovandi, et al., AIAFEX, Rio de Janeiro, 2003, math-ph/0303014, S. Joffily, "The Riemann Zeta Function and Vacuum Spectrum", Proceedings of Science, PoS (WC2004) 026, hep-th/0412217] where it was suggested a "potential scattering" Hilbert–Pólya conjecture, such that the nontrivial zeros of Riemann's zeta function could be put in one-to-one correspondence with the zeros of the s-wave Jost function for finite range potentials in the complex momenta plane, we extend our investigation to a relativistic S matrix for a Dirac particle scattering. We then present a description of the vacuum structure as being a dynamical system described by "virtual resonances", completely independent of the second quantization."

S. Joffily, "The Riemann zeta function and vacuum spectrum" (talk presented at the Fourth International Winter Conference on Mathematical Methods in Physics', Rio de Janeiro, 9-13 August 2004)

[abstract:] "A variant for the Hilbert and Pólya spectral interpretation of the Riemann zeta function is proposed. Instead of looking for a self-adjoint linear operator H, whose spectrum coincides with the Riemann zeta zeros, we look for the complex poles of the S-matrix that are mapped into the critical line in coincidence with the nontrivial Riemann zeroes. The associated quantum system, an infinity of "virtual resonances" described by the corresponding S-matrix poles, can be interpreted as the quantum vacuum. The distribution of energy levels differences associated to these resonances shows the same characteristic features of random matrix theory."

S. Joffily, "Jost function, prime numbers and Riemann zeta function" (preprint 03/03)

[abstract:] "The large complex zeros of the Jost function (poles of the S matrix) in the complex wave number-plane for s-wave scattering by truncated potentials are associated to the distribution of large prime numbers as well as to the asymptotic behavior of the imaginary parts of the zeros of the Riemann zeta function on the critical line. A variant of the Hilbert and Pólya conjecture is proposed and considerations about the Dirac sea as 'virtual resonances' are briefly discussed."

Articles by Alain Connes on noncommutative geometry and the zeta function

"We give a spectral interpretation of the critical zeros of the Riemann zeta function as an absorption spectrum, while eventual noncritical zeros appear as resonances." 

O. Bohigas, P. Leboeuf, M.-J. Sanchez, "On the distribution of the total energy of a system on non-interacting fermions: random matrix and semiclassical estimates" (preprint 06/98)

"A semiclassical formula describing, as a function of n, a non-universal behavior of the variance of the total energy starting at a critical number of particles is...obtained. It is illustrated with the particular case of single particle energies given by the imaginary part of the zeros of the Riemann zeta function on the critical line." 

O. Bohigas, P. Leboeuf, and M.-J. Sanchez, "Spectral spacing correlations for chaotic and disordered systems", Found. Phys. 31 (2001) 489-517

"New aspects of spectral fluctuations of (quantum) chaotic and diffusive systems are considered, namely autocorrelations of the spacing between consecutive levels or spacing autocovariances. They can be viewed as a discretized two point correlation function. Their behavior results from two different contributions. One corresponds to (universal) random matrix eigenvalue fluctuations, the other to diffusive or chaotic characteristics of the corresponding classical motion. A closed formula expressing spacing autocovariances in terms of classical dynamical zeta functions, including the Perron-Frobenius operator, is derived. It leads to a simple interpretation in terms of classical resonances. The theory is applied to zeros of the Riemann zeta function. A striking correspondence between the associated classical dynamical zeta functions and the Riemann zeta itself is found. This induces a resurgence phenomenon where the lowest Riemann zeros appear replicated an infinite number of times as resonances and sub-resonances in the spacing autocovariances. The theoretical results are confirmed by existing "data". The present work further extends the already well known semiclassical interpretation of properties of Riemann zeros."

P. Leboeuf, A. G. Monastra and O. Bohigas, "The Riemannium", Regular and Chaotic Dynamics 6 (2001) 205-210.

[abstract:] "The properties of a fictitious, fermionic, many-body system based on the complex zeros of the Riemann zeta function are studied. The imaginary part of the zeros are interpreted as mean-field single-particle energies, and one fills them up to a Fermi energy EF. The distribution of the total energy is shown to be non-Gaussian, asymmetric, and independent of EF in the limit EF -> infinity. The moments of the limit distribution are computed analytically. The autocorrelation function, the finite energy corrections, and a comparison with random matrix theory are also discussed."

P. Leboeuf and A.G. Monastra, "Quantum thermodynamic fluctuations of a chaotic Fermi-gas model", Nucl. Phys. A 724 (2003) 69-84

[abstract:] "We investigate the thermodynamics of a Fermi gas whose single-particle energy levels are given by the complex zeros of the Riemann zeta function. This is a model for a gas, and in particular for an atomic nucleus, with an underlying fully chaotic classical dynamics. The probability distributions of the quantum fluctuations of the grand potential and entropy of the gas are computed as a function of temperature and compared, with good agreement, with general predictions obtained from random matrix theory and periodic orbit theory (based on prime numbers). In each case the universal and non-universal regimes are identified."

T. Timberlake and J. Tucker, "Is there quantum chaos in the prime numbers?" (preprint 08/2007)

[abstract:] "We have computed the nearest neighbor spacing distribution, number variance, skewness, and excess for sequences of the first N prime numbers for various values of N. All four statistical measures clearly show a transition from random matrix statistics at small N toward Poisson statistics at large N. In addition, the number variance saturates at large length scales as is common for eigenvalue sequences. This data can be given a physical interpretation if the primes are thought of as eigenvalues of a quantum system whose classical dynamics is chaotic at low energy but regular at high energy. We discuss some difficulties with this interpretation in an attempt to clarify what kind of physical system might have the primes as its quantum eigenvalues."

C. Castro, "On two strategies towards the Riemann Hypothesis: Fractal Supersymmetric QM and a trace formula" (preprint 06/06)

[abstract:] "The Riemann Hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s_n =1/2+i lambda_n$. An improvement of our previous construction to prove the RH is presented by implementing the Hilbert-Pólya proposal and furnishing the Fractal Supersymmetric Quantum Mechanical (SUSY-QM) model whose spectrum reproduces the imaginary parts of the zeta zeros. We model the fractal fluctuations of the smooth Wu-Sprung potential (that capture the average level density of zeros) by recurring to a weighted superposition of Weierstrass functions $W(x,p,D)$ and where the summation has to be performed over all primes $p$ in order to recapture the connection between the distribution of zeta zeros and prime numbers. We proceed next with the construction of a smooth version of the fractal QM wave equation by writing an ordinary Schrödinger equation whose fluctuating potential (relative to the smooth Wu-Sprung potential) has the same functional form as the fluctuating part of the level density of zeros. The second approach to prove the RH relies on the existence of a continuous family of scaling-like operators involving the Gauss-Jacobi theta series. An explicit trace formula related to a superposition of eigenfunctions of these scaling-like operators is defined. If the trace relation is satisfied this could be another test of the Riemann Hypothesis."

C. Castro, A. Granik, and J. Mahecha, "On SUSY-QM, fractal strings and steps towards a proof of the Riemann hypothesis" (preprint 07/01)

[abstract:] "The steps towards a proof of Riemann's conjecture using spectral analysis are rigorously provided. We prove that the only zeros of the Riemann zeta-function are of the form $s = 1/2 + i\lambda_n$. A supersymmetric quantum mechanical model is proposed as an alternative way to prove the Riemann conjecture, inspired in the Hilbert-Pólya proposal; it uses an inverse scattering approach associated with a system of p-adic harmonic oscillators. An interpretation of the Riemann's fundamental relation Z(s) = Z(1 - s) as a duality relation, from one fractal string L to another dual fractal string L' is proposed."

C. Castro and J. Mahecha, "Fractal supersymmetric quantum mechanics, geometric probability and the Riemann Hypothesis", International Journal of Geometric Methods in Modern Physics 1 no. 6 (2004) 751-793

[abstract:] "The Riemann Hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s = 1/2 + i\lambda_{n}$. Earlier work on the RH based on Supersymmetric QM, whose potential was related to the Gauss-Jacobi theta series, allows to provide the proper framework to construct the well defined algorithm to compute the probability to find a zero (an infinity of zeros) in the critical line. Geometric Probability Theory furnishes the answer to the very difficult question whether the probability that the RH is true is indeed equal to unity or not. To test the validity of this Geometric Probabilistic framework to compute the probability if the RH is true, we apply it directly to the hyperbolic sine function sinh(s) case which obeys a trivial analog of the RH. Its zeros are equally spaced in the imaginary axis $s_n = 0 + in\pi$. The Geometric Probability to find a zero (and an infinity of zeros) in the imaginary axis is exactly unity. We proceed with a fractal supersymmetric quantum mechanical (SUSY-QM) model to implement the Hilbert-Pólya proposal to prove the RH by postulating a Hermitian operator that reproduces all the $\lambda_n$'s for its spectrum. Quantum inverse scattering methods related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) are applied to the analog of the fractal analog of the CBC (Comtet-Bandrauk-Campbell) formula in SUSY QM. It requires using suitable fractal derivatives and integrals of irrational order whose parameter $\beta$ is one-half the fractal dimension (D = 1.5) of the Weierstrass function. An ordinary SUSY-QM oscillator is also constructed whose eigenvalues are of the form $\lambda_n = n\pi$ and which coincide which the imaginary parts of the zeros of the function sinh(s). Finally, we discuss the relationship to the theory of 1/f noise."

C. Castro, "On the Riemann hypothesis, area quantization, Dirac operators, modularity, and renomalization group", International Journal of Geometric Methods in Modern Physics 7 (2010) 1 31

[abstract:] "Two methods to prove the Riemann Hypothesis are presented. One is based on the modular properties of $\Theta$ (theta) functions and the other on the Hilbert–Pólya proposal to find an operator whose spectrum reproduces the ordinates $\rho_n$ (imaginary parts) of the zeta zeros in the critical line: $s_n=1/2+i\rho n$. A detailed analysis of a one-dimensional Dirac-like operator with a potential $V(x)$ is given that reproduces the spectrum of energy levels $E_n = \rho_n$, when the boundary conditions $\Psi_E(x=-\infty)=\pm\Psi_E(x=+\infty) are imposed. Such potential $V(x)$ is derived implicitly from the relation $x=x(V)=\frac{\pi}{2}2 (dN(V)/dV), where the functional form of $N(V)$ is given by the full-fledged Riemann–von Mangoldt counting function of the zeta zeros, including the fluctuating as well as the $O(E^{-n})$ terms. The construction is also extended to self-adjoint Schroedinger operators. Crucial is the introduction of an energy-dependent cut-off function $\Lambda(E)$. Finally, the natural quantization of the phase space areas (associated to nonperiodic crystal-like structures) in integer multiples of $\pi$ follows from the Bohr–Sommerfeld quantization conditions of Quantum Mechanics. It allows to find a physical reasoning why the average density of the primes distribution for very large $x(O(\frac{1}{\log x}))$ has a one-to-one correspondence with the asymptotic limit of the $inverse$ average density of the zeta zeros in the critical line suggesting intriguing connections to the renormalization group program."

A. Odlyzko, "Primes, quantum chaos, and computers", from Number Theory (National Research Council, 1990) 35-46

N. Katz and P. Sarnak, "Zeroes of zeta functions and symmetry", Bulletin of the AMS, 36 (1999)

P. Sarnak, "Arithmetic quantum chaos", Israeli Mathematical Conference Proceedings 8 (1995) 183.

Physicists have long studied spectra of Schrödinger operators and random matrices, thanks to the implications for quantum mechanics. Analogously number theorists and geometers have investigated the statistics of spectra of Laplacians on Riemannian manifolds associated with arithmetic groups. Sarnak calls this "arithmetic quantum chaos." Equivalently one studies the zeros of Selberg zeta functions. Parallels with the statistics of the zeros of the Riemann zeta function have been evident for some time.

P.E. Cartier, B. Julia, P. Moussa and P. Vanhove (eds.), Frontiers in Number Theory, Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems (Springer, 2006)

[publisher's description:] "This book presents pedagogical contributions on selected topics relating Number Theory, Theoretical Physics and Geometry. The parts are composed of long self-contained pedagogical lectures followed by shorter contributions on specific subjects organized by theme. Most courses and short contributions go up to the recent developments in the fields; some of them follow their author's original viewpoints. There are contributions on Random Matrix Theory, Quantum Chaos, Non-commutative Geometry, Zeta functions, and Dynamical Systems. The chapters of this book are extended versions of lectures given at a meeting entitled Number Theory, Physics and Geometry, held at Les Houches in March 2003."

E. Bogomolny, "Quantum and arithmetical chaos" (preprint 12/03, based on lectures given at Les Houches School "Frontiers in Number Theory, Physics and Geometry", March 2003)

[abstract:] "The lectures are centered around three selected topics of quantum chaos: the Selberg trace formula, the two-point spectral correlation functions of Riemann zeta function zeros, and of the Laplace-Beltrami operator for the modular group. The lectures cover a wide range of quantum chaos applications and can serve as a non-formal introduction to mathematical methods of quantum chaos."

R. Aurich, F. Scheffler and F. Steiner, "On the subtleties of arithmetical quantum chaos", Physical Review E 51 (1995) 4173.

J. Bolte, "Periodic orbits in arithmetical chaos" (preprint, 1992)

[abstract:] "Length spectra of periodic orbits are investigated for some chaotic dynamical systems whose quantum energy spectra show unexpected statistical properties and for which the notion of arithmetical chaos has been introduced recently. These systems are defined as the unconstrained motions of particles on two dimensional surfaces of constant negative curvature whose fundamental groups are given by number theoretical statements (arithmetic Fuchsian groups). It is shown that the mean multiplicity of lengths $l$ of periodic orbits grows asymptotically like $c\cdot e^{l/2}/l$, $l\rto \infty$. Moreover, the constant $c$ (depending on the arithmetic group) is determined."

J. Bolte, "Some studies on arithmetical chaos in classical and quantum mechanics" (preprint 05/93)

[Abstract:] "Several aspects of classical and quantum mechanics applied to a class of strongly chaotic systems are studied. These consist of single particles moving without external forces on surfaces of constant negative Gaussian curvature whose corresponding fundamental groups are supplied with an arithmetic structure. It is shown that the arithmetic features of the considered systems lead to exceptional properties of the corresponding spectra of lengths of periodic orbits. The most significant one is an exponential growth of degeneracies in these length spectra. Furthermore, the arithmetical systems are distinguished by a structure that appears as a generalization of geometric symmetries. These pseudosymmetries occur in the quantization of the classical arithmetic systems as Hecke operators, which form an infinite algebra of self-adjoint operators commuting with the Hamiltonian. The statistical properties of quantum energies in the arithmetical have previously been identified as exceptional. They do not fit into the general scheme of random matrix theory. It is shown with the help of a simplified model for the spectral form factor how the spectral statistics in arithmetic quantum chaos can be understood by the properties of the corresponding classical length spectra. A decisive is played by the exponentially increasing multiplicities of lengths. The model developed for the level spacings distribution and for the number variance is compared to the corresponding quantities obtained from quantum energies for a specific arithmetical system."

J. Main, V. Mandelshtam, and H. Taylor, "Periodic orbit quantization by harmonic inversion of Gutzwiller's recurrence function", Physical Review Letters 79 no. 5 (1997)

"Semiclassical eigenenergies and resonances are obtained from classical periodic orbits by harmonic inversion of Gutzwiller's semiclassical recurrence function, i.e., the trace of the propagator. Applications to the chaotic three disk scattering system and, as a mathematical model, to the Riemann zeta function demonstrate the power of the technique. The method does not depend on the existence of a symbolic code and might be a tool for a semiclassical quantization of systems with nonhyperbolic or mixed regular-chaotic dynamics as well."

J. Main, V.A. Mandelshtam, G. Wunner and H.S. Taylor, "Riemann zeros and periodic orbit quantization by harmonic inversion" (preprint 09/97)

[abstract:] "In formal analogy with Gutzwiller's semiclassical trace formula, the density of zeros of the Riemann zeta function zeta(z=1/2-iw) can be written as a non-convergent series rho(w)=-pi^{-1} sum_p sum_{m=1}^infty ln(p)p^{-m/2} cos(wm ln(p)) with p running over the prime numbers. We obtain zeros and poles of the zeta function by harmonic inversion of the time signal which is a Fourier transform of rho(w). More than 2500 zeros have been calculated to about 12 digit precision as eigenvalues of small matrices using the method of filter-diagonalization. Due to formal analogy of the zeta function with Gutzwiller's periodic orbit trace formula, the method can be applied to the latter to accurately calculate individual semiclassical eigenenergies and resonance poles for classically chaotic systems. The periodic orbit quantization is demonstrated on the three disk scattering system as a physical example."

J. Main, P.A. Dando, Dz. Belkic and H S Taylor, "Decimation and harmonic inversion of periodic orbit signals", J. Phys.A: Math. Gen. 33 (2000) 1247-1263.

[excerpts:] "Introduction. The semiclassical quantization of systems with an underlying chaotic classical dynamics is a nontrivial problem due to the fact that Gutzwiller's trace formula [1, 2] does not usually converge in those regions where the eigenenergies or resonances are located. Various techniques have been developed to circumvent the convergence problem of periodic orbit theory. Examples are the cycle expansion technique [3], the Riemann-Siegel-type formula and pseudo-orbit expansions [4], surface of section techniques [5], and a quantization rule based on a semiclassical approximation to the spectral staircase [6]. These techniques have proven to be very efficient for systems with special properties, e.g., the cycle expansion for hyperbolic systems with an existing symbolic dynamics, while the other methods mentioned have been used for the calculation of bound spectra.


In section 5 we present and compare results for the three-disc scattering system as a physical example and the zeros of the Riemann zeta function as a mathematical model for periodic orbit quantization. Some concluding remarks are given in section 6."

more papers by J. Main, et. al.

E. Doron, "Do spectral trace formulae converge?" (preprint 02/95)

[abstract:] "We evaluate the Gutzwiller trace formula for the level density of classically chaotic systems by considering the level density in a bounded energy range and truncating its Fourier integral. This results in a limiting procedure which comprises a convergent semiclassical approximation to a well defined spectral quantity at each stage. We test this result on the spectrum of zeros of the Riemann zeta function, obtaining increasingly good approximations to the level density. The Fourier approach also explains the origin of the convergence problems encountered by the orbit truncation scheme."

A.V. Andreev, O. Agam, B.D. Simons, B.L. Altschuler, "Quantum chaos, irreversible classical dynamics, and random matrix theory", Physical Review Letters 76 (1996) 3497

[abstract:] "The Bohigas-Giannoni-Schmidt conjecture stating that the statistical spectral properties of systems which are chaotic in their classical limit coincide with random matrix theory is proved. For this purpose a new semiclassical field theory for individual chaotic systems is constructed in the framework of the non--linear $\sigma$-model. The low lying modes are shown to be associated with the Perron-Frobenius spectrum of the underlying irreversible classical dynamics. It is shown that the existence of a gap in the Perron-Frobenius spectrum results in a RMT behavior. Moreover, our formalism offers a way of calculating system specific corrections beyond RMT."

O. Agam, A.V. Andreev, B.L. Altshuler, "Relations between quantum and classical spectral determinants (zeta-functions)" (preprint 02/96)

"We demonstrate that beyond the universal regime correlators of quantum spectral determinants $\Delta(\epsilon)=\det (\epsilon-\hat{H})$ of chaotic systems, defined through an averaging over a wide energy interval, are determined by the underlying classical dynamics through the spectral determinant $1/Z(z)=\det (z- {\cal L})$,  where $e^{-{\cal L}t}$ is the Perron-Frobenius operator. Application of these results to the Riemann zeta function, allows us to conjecture new relations satisfied by this function."

M. Rubinstein, "Evidence for a spectral interpretation of zeros of L-functions" (Princeton University Ph.D. thesis, 1998)

"...provides additional theoretical and numerical evidence connecting the zeros of L-functions (generalization of Riemann Zeta) to eigenvalues of operators from the classical compact groups (unitary, orthogonal, unitary symplectic)."

S.N. Evangelou and D.E. Katsanos, "Quantum correlations from Brownian diffusion of chaotic level-spacings", Physics Letters A 334 nos.5-6 (2005) 331-336

[abstract:] "Quantum chaos is linked to Brownian diffusion of the underlying quantum energy level-spacing sequences. The level-spacings viewed as functions of their order execute random walks which imply uncorrelated random increments of the level-spacings while the integrability to chaos transition becomes a change from Poisson to Gauss statistics for the level-spacing increments. This universal nature of quantum chaotic spectral correlations is numerically demonstrated for eigenvalues from random tight binding lattices and for zeros of the Riemann zeta function."

W.T. Lu and S. Sridhar, "Correlations among the Riemann zeros: Invariance, resurgence, prophecy and self-duality" (preprint 05/04)

[abstract:] "We present a conjecture describing new long range correlations among the Riemann zeros leading to 3 principal features: (i) The spectral auto-correlation is invariant with respect to the averaging window. (ii) Resurgence occurs wherein the lowest zeros appear in all auto-correlations. (iii) Suitably defined correlations lead to predictions (prophecy) of new zeros. This conjecture is supported by analytical arguments and confirmed by numerical calculations using 1022 zeros computed by Odlyzko. The results lead to a self-duality of the Riemann spectrum similar to the quantum-classical duality observed in billiards."

A. Kuznetsov, "Nontrivial zeros of the Riemann zeta function as the limit of eigenvalues of nonsymmetric matrices" (preprint 2006)

[abstract:] "We construct a family of $2n \times 2n$ matrices $B_{2n}$, such that the spectrum of $\sigma(B_{2n})$ converges to the set of nontrivial zeros of the Riemann zeta function $\zeta(s)$. The coefficients of these matrices are given explicitly as finite sums of Bernoulli numbers."

Y.-H. He, V. Jejjala, D. Minic, "Eigenvalue density, Li's positivity, and the critical strip" (preprint 03/2009)

[abstract:] "We rewrite the zero-counting formula within the critical strip of the Riemann zeta function as a cumulative density distribution; this subsequently allows us to derive an integral expression for the Li coefficients associated with the Riemann xi-function and, in particular, indicate that their positivity criterion is obeyed, whereby entailing the criticality of the non-trivial zeros. We also offer a physical interpretation of the result and discuss the Hilbert-Polya approach."

A. Lipniacka and B.M. Dit Latour, "A quantum mechanical well and a derivation of a $\pi^2$ formula" (preprint 11/2017)

[abstract:] "Quantum particle bound in an infinite, one-dimensional square potential well is one of the problems in Quantum Mechanics (QM) that most of the textbooks start from. There, calculating an allowed energy spectrum for an arbitrary wave function often involves Riemann zeta function resulting in a $\pi$ series. In this work, two "$\pi$ formulas" are derived when calculating a spectrum of possible outcomes of the momentum measurement for a particle confined in such a well, the series, $\frac{\pi^2}{8} = \sum_{k=1}^{k=\infty} \frac{1}{(2k-1)^2}$, and the integral $\int_{-\infty}^{\infty} \frac{sin^2 x}{x^2} dx =\pi$. The spectrum of the momentum operator appears to peak on classically allowed momentum values only for the states with even quantum number. The present article is inspired by another quantum mechanical derivation of $\pi$ formula in \cite{wallys}."

P. Riot and A. Le Méhauté, "From the arrow of time in Badiali's quantum approach to the dynamic meaning of Riemann's hypothesis" (preprint 10/2017)

[abstract:] "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."

P. Braun and D. Waltner, "New approach to periodic orbit theory of spectral correlations" (preprint 09/2018)

[abstract:] "The existing periodic orbit theory of spectral correlations for classically chaotic systems relies on the Riemann–Siegel-like representation of the spectral determinants which is still largely hypothetical. We suggest a simpler derivation using analytic continuation of the periodic-orbit expansion of the pertinent generating function from the convergence border to physically important real values of its arguments. As examples we consider chaotic systems without time reversal as well as the Riemann zeta function and Dirichlet $L$-functions zeros."

R.A. Ribeiro Correia de Sousa, "Prime numbers: A particle in a box and the complex wave model", Journal of Mathematics Research 7 (4) (2015)

[abstract:] "Euler's formula establishes the relationship between the trigonometric function and the exponential function. In doing so unifies two waves, a real and an imaginary one, that propagate through the Complex number set, establishing relation between integer numbers. A complex wave, if anchored by zero and by a defined integer number $N$, only can assume certain oscillation modes. The first mode of oscillation corresponds always to a $N$ prime number and the other modes to its multiples.

\(\psi (x)=x e^{i\left(\frac{n \pi }{N}x\right)}\)

Under the above described conditions, these waves and their admissible oscillation modes allows for primality testing of integer numbers, the deduction of a new formula $\pi(x)$ for counting prime numbers and the identification of patterns in the prime numbers distribution with computing time gains in the calculations. In this article, four theorems and one rule of factorizing are put forward with consequences for prime number signaling, counting and distribution. Furthermore, it is establish the relationship between this complex wave with a time independent semi-classical harmonic oscillator, in which the spectrum of the allowed energy levels are always only prime numbers. Thus, it is affirmative the reply to the question if the prime numbers distribution is related to the energy levels of a physical system."

Matti Pitkänen, Quantum TGD and how to prove Riemann hypothesis (3/2/2001)

"During last month further ideas about Riemann hypothesis have emerged and have led to further sharpening of Riemann hypothesis and to p-adic particle physicist's articulation for what it is to be zero of Riemann Zeta and to the idea that Riemann hypothesis reduces to superconformal invariance of the physical system involved. One can verify Hilbert-Pólya hypothesis on basis of the physical picture obtained. This means an explicit construction of the differential operator having the moduli squared of the zeros of Riemann Zeta as eigenvalues. This operator is product of two operators which are Hermitian conjugates of each other and have zeros of Riemann Zeta as their eigenvalues. The facts that x corresponds to the real part of conformal weight in this model and that one has x = n/2 for the operators appearing in the representations of Super Virasoro, suggest that x = n/2 is indeed the only possible value of x for the zeros of Riemann zeta both in real and p-adic context. Hence Riemann hypothesis would indeed reduce to superconformal invariance."

M. Pitkänen, "A further step in the proof of Riemann Hypothesis"


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