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.