scattering theory and number theory

S. Torquato, G. Zhang and M. de Courcy-Ireland, "Uncovering multiscale order in the prime numbers via scattering" (preprint 02/2018)

[abstract:] "The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which we call effectively limit-periodic. In particular, the primes in this regime are hyperuniform. This is shown analytically using the structure factor $S(k)$, proportional to the scattering intensity from a many-particle system. Remarkably, the structure factor for primes is characterized by dense Bragg peaks, like a quasicrystal, but positioned at certain rational wavenumbers, like a limit-periodic point pattern. We identify a transition between ordered and disordered prime regimes that depends on the intervals studied. Our analysis leads to an algorithm that enables one to predict primes with high accuracy. Effective limit-periodicity deserves future investigation in physics, independent of its link to the primes."

Y. Abe, "Application of abelian holonomy formalism to the elementary theory of numbers" (preprint 06/2010)

[abstract:] "We consider an abelian holonomy operator in two-dimensional conformal field theory with zero-mode contributions. The analysis is made possible by use of a geometric-quantization scheme for abelian Chern-Simons theory on $S^1 \times S^1 \times {\bf R}$. We find that a purely zero-mode part of the holonomy operator can be expressed in terms of Riemann's zeta function. We also show that a generalization of linking numbers can be obtained in terms of the vacuum expectation values of the zero-mode holonomy operators. Inspired by mathematical analogies between linking numbers and Legendre symbols, we then apply these results to a space of ${\bf F}_p = {\bf Z}/ p {\bf Z}$ where $p$ is an odd prime number. This enables us to calculate "scattering amplitudes" of identical odd primes in the holonomy formalism. In this framework, the Riemann hypothesis can be interpreted by means of a physically obvious fact, i.e., there is no notion of "scattering" for a single-particle system. Abelian gauge theories described by the zero-mode holonomy operators will be useful for studies on quantum aspects of topology and number theory."

J.-F. Burnol, "An adelic causality problem related to abelian L-functions" J. Number Theory 87 Série I (2001) 423-428

In this paper, Burnol uses a Lax-Phillips scattering framework to reveal "a natural formulation of the Riemann Hypothesis, simultaneously for all L-functions, as a property of causality."

[excerpts:] "The study of connections between the Riemann zeta function and scattering theory is at least thirty years old. In particular the Faddeev-Pavlov study of scattering for automorphic functions [FP] further developed by Lax and Phillips in [LP] has attracted widespread attention. In their approach the scattering matrix is directly related to the values taken by the Riemann zeta function on the line Re[s] = 1, and the Riemann Hypothesis itself is equivalent to some decay properties of scattering waves. Another well-known instance is the approach of De Branges ([DB1], [DB2]) within the theory of Hilbert spaces of entire functions, also related to scattering. Conrey and Li have recently pointed out some difficulties of this approach:

The connection between our scattering process and the Riemann zeta the following: each 'bad' zero (Re[s] > 1/2) appears as a pole of the scattering operator, where there should be none, if the process was causal. But if the Riemann Hypothesis holds, then the scattering itself is of a trivial nature, and says absolutely nothing on the zeros on the critical line...Our sole motivation in formulating the Riemann Hypothesis in a novel manner is the hope that creators of other tools, of a deeper nature than those used here, would incorporate the gained insight in their design constraints."

[FP] L.D Faddeev and B.S. Pavlov, "Scattering theory and automorphic functions", Seminar of Steklov Mathematical Institute of Leningrad 27 (1972) 161-193.

[LP] P. Lax and R.S. Phillips, Scattering Theory (1st edition, 1967) revised edition, Pure and Applied Mathematics 26 (Academic Press, 1989)

[DB1] L. De Branges, "The Riemann hypothesis for Hilbert spaces of entire functions", Bulletin of the American Mathematical Society (New Series) 15 no. 1 (1986) 1-17.

[DB2] L. De Branges, "A conjecture which implies the Riemann hypothesis", Journal of Functional Analysis 121 no. 1 (1994) 117-184.

[CL] J.B Conrey and X.-J. Li, "A note on some positivity conditions related to zeta- and L-functions", American Institute of Mathematics preprint series (1998)

J.-F. Burnol, "Scattering for time series with an application to the zeta function of an algebraic curve" (preprint, 11/99)

[abstract:] "I explain how the Lax-Phillips theory can be applied to a purely innovating time series and compute the corresponding scattering function. I then associate such a time series to an algebraic curve (of genus at least 1) over a finite field and show that the Riemann Hypothesis (proven long ago) holds if and only if the scattering is causal (this causality is not independently established, though)."

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, "Scattering on the p-adic field and a trace formula" Int. Math. Res. Not. 2 (2000) 57-70

[abstract] "I apply the set-up of Lax-Phillips Scattering Theory to a non-archimedean local field. It is possible to choose the outgoing space and the incoming space to be Fourier transforms of each other. Key elements of the Lax-Phillips theory are seen to make sense and to have the expected interrelations: the scattering matrix S, the projection K to the interacting space, the contraction semi-group Z and the time delay operator T. The scattering matrix is causal, its analytic continuation has the expected poles and zeros, and its phase derivative is the (non-negative) spectral function of T, which is also the restriction to the diagonal of the kernel of K. The contraction semi-group Z is related to S (and T) through a trace formula. Introducing an odd-even grading on the interacting space allows to express the Weil local explicit formula in terms of a "supertrace". I also apply my methods to the evaluation of a trace considered by Connes."

J.-F. Burnol, "On Fourier and zeta(s)" (habilitation thesis, submitted to Forum Mathematicum)

[abstract] "We study some of the interactions between the Fourier Transform and the Riemann zeta function (and Dirichlet-Dedekind-Hecke-Tate L-functions)."

This involves in-depth discussion of such physics-related topics as scattering, causality, the Kramers-Wannier duality relation, and the Hilbert-Pólya idea.

[excerpt] "But we do believe that some sort of a much more hidden thing exists, a Kramers-Wannier like duality exchanging the low temperature phase with a single hot temperature phase, not number-theoretical. If this were really the case, some universal properties would hold across all phases, reflecting the universality exemplified by the GUE hypothesis. Of course the hot phase is then expected to be somehow related with quantities arising in the study of random matrices. In the picture from Theorem 7.2, $\Lambda$ seems to play the role of an inverse temperature (coupling constant).

We expect that if such a duality did reign on our space it would interact in such a manner with the renormalization group flow that this would give birth to scattering processes. Indeed the duality could be used to compare incoming to outgoing (classical) states. Perhaps the constraints related with this interaction would result in a property of causality equivalent to the Riemann Hypothesis."

M.A. Semenov-Tian-Shansky, "Harmonic analysis on Riemannian symmetric spaces of negative curvature and scattering theory", Math. USSR Izvestija 10 (1976) 535-563.

P.D. Lax and R.S. Phillips, "Scattering theory for automorphic functions", Bulletin of the American Mathematical Society 2 (2) (1980) 261- 295.

[abstract:] "This paper is an expository account of our 1976 monograph on Scattering theory for automorphic functions. Several improvements have been incorporated: a more direct proof of the meromorphic character of the Eisenstein series, an explicit formula for the translation representations and a simpler derivation of the spectral representations. our hyperbolic approach to the Selberg trace formula is also included."

M.C. Gutzwiller, "Stochastic behavior in quantum scattering", Physica D: Nonlinear Phenomena 7 (1983) 341-355

[abstract:] "A 2-dimensional smooth orientable, but not compact space of constant negative curvature with the topology of a torus is investigated. It contains an open end, i.e. an exceptional point at infinite distance, through which a particle or a wave can enter or leave, as in the exponential horn of certain antennas or loud-speakers. In the Poincaré model of hyperbolic geometry, the solutions of Schrödinger's equation for the reflection of a particle which enters through the horn are easily constructed. The scattering phase shift as a function of the momentum is essentially given by the phase angle of Riemann's zeta function on the imaginary axis, at a distance of from the famous critical line. This phase shift shows all the features of chaos, namely the ability to mimick any given smooth function, and great difficulty in its effective numerical computation. A plot shows the close connection with the zeros of Riemann's zeta function for low values of the momentum (quantum regime) which gets lost only at exceedingly large momenta (classical regime?) Some generalizations of this approach to chaos are mentioned."

general notes on the number theoretic content of Gutzwiller's publications involving further remarks on scattering on surfaces of constant negative curvature, the Selberg Trace Formula, Voronin's Universality Theorem concerning the Riemann zeta function, etc.

D.M. Wardlaw and W. Jaworski, "Time delay, resonances, Riemann zeros and chaos in a model quantum scattering system", Journal of Physics A 22 (1989) 3561-3575.

[Abstract:] "The quantum treatment of an intrinsically chaotic model scattering system originally studied by Gutzwiller (1983) is extended to include the time delay and to make explicit the zeros of the Riemann zeta function in the mathematical expressions for the scattering matrix S and the time delay. The system consists of a particle moving on a two-dimensional surface of constant negative curvature. The scattering in this unusual system is dominated by resonances associated with poles of S in the complex momentum plane, the real parts of these poles are one-half of the imaginary parts of the Riemann zeros. The resonances have a constant width but their average spacing varies with the momentum. The focal point is the manifestation of chaotic behaviour in the time delay. Features considered in this regard include: the momentum dependence of the time delay in regions of overlapping and isolated resonances, the decomposition of the time delay into an average (dynamical) component and a fluctuating (chaotic) component, and characterisation of the fluctuating component by its autocorrelation function."

P.G.O. Freund, "Scattering on p-adic on adelic symmetric spaces", Physics Letters B 257 (1991) 119-124

[abstract:] "Explicit S-matrices are constructed for scattering on p-adic hyperbolic planes. Combining these with the known S-matrix on the real hyperbolic plane, an adelic S-matrix is obtained. It has poles at the nontrivial zeros of the Riemann zeta-function, and is closely related to scattering on the modular domain of the real hyperbolic plane. Generalizations of this work and their possible arithmetic relevance are outlined."

J.B Conrey and X.-J. Li, "A note on some positivity conditions related to zeta- and L-functions", American Institute of Mathematics preprint series (1998)

[abstract:] "In his recent work, Louis de Branges proposed an approach to the generalized Riemann hypothesis, that is, the hypothesis that not only the Riemann zeta function but also all the Dirichlet L-functions with primitive Dirichlet characters have their nontrivial zeros lying on the critical line. In a paper, de Branges mentioned that his approach to the generalized Riemann hypothesis using Hilbert spaces of entire functions is related to the Lax-Phillips theory of scattering. Peter Lax and Ralph Phillips explained in their book the difficulty of approaching the Riemann hypothesis by using the scattering theory. In this note, we shall give examples showing that de Branges' positivity conditions, which imply the generalized Riemann hypothesis, are not satisfied by defining functions of reproducing kernel Hilbert spaces associated with the Riemann zeta function and the Dirichlet L-function L(s;4)."

S.-H. Tang and M. Zworski, "Resonance expansions of scattered waves", Commun. Pure Appl. Math. 53 (2000) 1305-1334

[abstract:] "The purpose of this paper is to describe expansions of solutions to the wave equation on Rn with a compactly supported perturbation present. We show that under a separation condition on resonances, the solutions can be expanded in terms of resonances close to the real axis, modulo an error rapidly decaying in time. To avoid the discussion of particular aspects of potential, gravitational or obstacle scattering, the results are stated using the abstract 'black box' formalism of Sjöstrand and the second author."

[excerpt, p.2:] "Two hyperbolic examples: the modular surface and the hyperbolic cylinder are shown to admit expansion of waves in terms of resonances. However, even there, the strongest expected version of the expansions is not available as it would require too much knowledge about the Riemann zeta function (simplicity and lower bounds on derivatives)..."

P. Lévay, "Chaotic Aharonov-Bohm scattering on surfaces of constant negative curvature", Journal of Physics A 33 (2000) 4129-4141.

[Abstract:] "A topological model of the Aharonov-Bohm scattering is presented, where the usual set-up is modelled by a genus-one Riemann surface with two cusps, i.e. leaks infinitely far away. This constant negative-curvature surface is uniformized by the Hecke congruence subgroup $\Gamma_{0}(11)$ of the modular group. The fluxes through the holes are described by the even Dirichlet character for $\Gamma_{0}(11)$. The scattering matrix having only off-diagonal elements (no reflection) is calculated. The fluctuating part of the off-diagonal entries shows a non-trivial dependence on the fluxes as well. The scattering resonances are related to the non-trivial zeros of a Dirichlet L-function. The chaotic nature of the scattering is related to the distribution of primes in arithmetical progressions."

P. Lévay, "Chaotic Aharonov-Bohm scattering on surfaces of constant negative curvature", Journal of Physics A 33 (2000) 4129-4141.

[Abstract:] "A topological model of the Aharonov-Bohm scattering is presented, where the usual set-up is modelled by a genus-one Riemann surface with two cusps, i.e. leaks infinitely far away. This constant negative-curvature surface is uniformized by the Hecke congruence subgroup $\Gamma_{0}(11)$ of the modular group. The fluxes through the holes are described by the even Dirichlet character for $\Gamma_{0}(11)$. The scattering matrix having only off-diagonal elements (no reflection) is calculated. The fluctuating part of the off-diagonal entries shows a non-trivial dependence on the fluxes as well. The scattering resonances are related to the non-trivial zeros of a Dirichlet L-function. The chaotic nature of the scattering is related to the distribution of primes in arithmetical progressions."

M. Pigli, "Adelic integrable systems" (preprint 07/03)

[abstract:] "Incorporating the zonal spherical function (zsf) problems on real and p-adic hyperbolic planes into a Zakharov-Shabat integrable system setting, we find a wide class of integrable evolutions which respect the number-theoretic properties of the zsf problem. This means that at all times these real and p-adic systems can be unified into an adelic system with an S-matrix which involves (Dirichlet, Langlands, Shimura...) L-functions."

[introduction:] "Scattering theory on real [1] and p-adic [2] symmetric spaces can be unified in an adelic context. This had the virtue of producing S-matrices involving the Riemann zeta function and of throwing new light on earlier work [4] concerning scattering on the noncompact finite-area fundamental domain of SL(2,Z) on the Real hyperbolic plane $H_{\infty}$.

The real hyperbolic plane is a smooth manifold and as such quantum mechanics on $H_{\infty}$ involves a second order Schrodinger differential equation. By contrast the p-adic hyperbolic planes Hp are discrete spaces (trees), and the corresponding Schrodinger equations are second order difference equations. The Jost functions, and therefore the S-matrices from all these local problems combine in adelic products, which then involve the Riemann zeta function [2].

At a given time consider all these ("S-wave") scattering problems and then let all of them undergo an integrable time evolution. In general such an evolution need not respect the number-theoretic endowment of the initial problem. In other words, even though at the initial time the real and p-adic scattering problems assembled into an interesting adelic scattering problem, at later times this need no longer be so. We want to explore here the conditions under which the integrable evolution respects adelizability and to see what kind of scattering problems can be obtained this way at later times. Specifically, we will incorporate the initial scattering problem into a Zakharov-Shabat (ZS) system and follow its integrable evolution. For the p-adic problems, time has to be discrete and for adelic purposes time then has to be discrete in the real problem as well. We will see that along with the Riemann zeta function involved in the adelic problem at the initial time, various (Dirichlet, Langlands, Shimura,...) L-functions [5] appear at later times."

[1] M.A. Olshanetsky and A.M. Perelomov, Phys. Rep. 94 (1984) 313; R.F. Wehrhahn, Rev. Math. Phys. 6 (1994) 1339
[2] P.G.O. Freund, Phys. Lett. B 257 (1919) 119; L. Brekke and P.G.O. Freund, Phys. Rep. 233 (1993) 1
[3] L.O. Chekhov, "L-functions in scattering on p-adic multiloop surfaces", J. Math. Phys. 36 (1995) 414
[4] L.D. Faddeev and B.S. Pavlov, Sem. LOMI 27 (1972) 161; P.D. Lax and R.S. Phillips, Scattering Theory for Automorphic Functions (Princeton Univ. Press, 1976)
[5] S. Gelbart and F. Shahidi, Analytic Properties of Automorphic L-Functions (Academic, 1988)

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, "Jost function, prime numbers and Riemann zeta function" (preprint 03/2003)

[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."

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."

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 $\zeta(s)$."

G. Chalmers, "Comment on the Riemann hypothesis" (preprint 03/05)

[abstract:] "The Riemann hypothesis is identified with zeros of ${\cal N}=4$ supersymmetric gauge theory four-point amplitude. The zeros of the $\zeta(s)$ function are identified with the complex dimension of the spacetime, or the dimension of the toroidal compactification. A sequence of dimensions are identified in order to map the zeros of the amplitude to the Riemann hypothesis."

A. LeClair, "Interacting Bose and Fermi gases in low dimensions and the Riemann hypothesis" (preprint, 11/2006)

[abstract:] "We apply the S-matrix based finite temperature formalism we recently developed to non-relativistic Bose and Fermi gases in 1+1 and 2+1 dimensions. In the 2+1 dimensional case, the free energy is given in terms of Roger's dilogarithm in a way analagous to the relativistic 1+1 dimensional case. The 1-d fermionic case with a quasi-periodic 2-body potential provides a physical framework for understanding the Riemann hypothesis."

The follow purportedly reduces the Riemann Hypothesis to an inverse (quantum) scattering problem, and (despite the humility of the title), claims to contain a proof of the RH.

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 zeroes 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 witha 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, "A fractal SUSY-QM model and the Riemann hypothesis" (preprint 06/03)

[abstract:] "The Riemann hypothesis (RH) states that the nontrivial zeros of the Riemann zeta-function are of the form $s = 1/2 + i\lambda_n$. Hilbert-Pólya argued that if a Hermitian operator exists whose eigenvalues are the imaginary parts of the zeta zeros, $\lambda_n$, then the RH is true. In this paper a fractal supersymmetric quantum mechanical (SUSY-QM) model is proposed to prove the RH. It is based on a quantum inverse scattering method related to a fractal potential given by a Weierstrass function (continuous but nowhere differentiable) that is present in 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 of the Weierstrass function. An ordinary SUSY-QM oscillator is constructed whose eigenvalues are of the form $\lambda_n = n\pi$, and which coincide with the imaginary parts of the zeros of the funciton sin(iz). This sine function obeys a trivial analog of the RH. A review of our earlier proof of the RH based on a SUSY QM model whose potential is related ot the Gauss-Jacobi theta series is also included. The spectrum is given by s(1 - s) which is real in the critical line (location of the nontrivial zeros) and in the real axis (location of the trivial zeros)."

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 and tachyons in dual string scattering amplitudes", International Journal of Geometric Methods in Modern Physics 3 no. 2 (2006) 187-199

[abstract:] "It is the purpose of this work to pursue a novel physical interpretation of the nontrivial Riemann zeta zeros and prove why the location of these zeros $z_n = 1/2+i\lambda_n$ corresponds physically to tachyonic-resonances/tachyonic-condensates, originating from the scattering of two on-shell tachyons in bosonic string theory. Namely, we prove that if there were nontrivial zeta zeros (violating the Riemann hypothesis) outside the critical line Real $z = 1/2$ (but inside the critical strip), these putative zeros do not correspond to any poles of the bosonic open string scattering (Veneziano) amplitude $A(s,t,u)$. The physical relevance of tachyonic-resonances/tachyonic-condensates in bosonic string theory, establishes an important connection between string theory and the Riemann Hypothesis. In addition, one has also a geometrical interpretation of the zeta zeros in the critical line in terms of very special (degenerate) triangular configurations in the upper-part of the complex plane."

The following appears to be related to the work of Castro, et. al., insofar as it involves inverse scattering:

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."

A. Wirzba, Zeta functions in scattering problems evaluated on the real wave number axis"

[abstract:] "Using the two-dimensional 3-disk system (in the A1 representation) we compare the cluster phase shifts of the exact quantum mechanical problem with the corresponding cluster phase shift of the following semi-classical zeta functions: (a) the Gutzwiller-Voros zeta function, (b) the dynamical zeta function and (c) the quasi-classical determinant. Furthermore we show results for the squared moduli of the quantum mechanical and semi-classical determinants on the real wave number axis. As the cluster phase shifts are in principle measurable quantities, even experimentalists can now decide which of the three choices is the best."

S.C. Woon, "Analytic continuation of Bernoulli numbers, a new formula for the Riemann zeta function, and the phenomenon of scattering of zeros" (preprint 05/97)

"The new concept of dynamics of the zeros of analytic continued polynomials is introduced, and an interesting phenomenon of 'scatterings' of the zeros of [the Bernoulli polynomials] Bs(z) is observed"

Animations of the above phenomenon can be viewed here

R.K. Bhaduri, Avinash Khare, and J. Law, "Phase of the Riemann zeta function and the inverted harmonic oscillator", Physical Review E 52 no. 1 (1995) 486-491

[abstract] The Argand diagram is used to display some characteristics of the characteristics of the Riemann zeta function. The zeros of the zeta function on the complex plane give rise to an infinite sequence of closed loops, all passing through the origin of the diagram. The behaviour of the phase of he zeta function on and off the line of zeros is studied. Up to some distance from the line of the complex zeros, the phase angle is shown to still retain their memory. 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."

R.K. Bhaduri, Avinash Khare, S.M. Reimann, and E.L. Tomusiak, "The Riemann zeta function and the inverted harmonic oscillator", Annals of Physics 254 no. 1 (1997)

[abstract] The Riemann zeta function has phase jumps of $\pi$ every time it changes sign as the parameter t in the complex argument s = 1/2 + it is varied. We show analytically that as the real part of the argument is increased to $\rho$ > 1/2, the memory of the zeros fades only gradually through a Lorentzian smoothing of the density of the zeros. The corresponding trace formula, for $\rho$ >> 1, is of the same form as that generated by a one-dimensional harmonic oscillator in one direction, along with an inverted oscillator in the transverse direction. It is pointed out that Lorentzian smoothing of the level density for more general dynamical systems may be done similarly. The Gutzwiller trace formula for the simple saddle plus oscillator model is obtained analytically, and is found to agree with the quantum result


The Riemann zeta function zeta(s) of the complex variable s = $\rho$ + it has an infinite number of zeros on the half-line $\rho$ = 1/2 [1]. Along this line, as a function of t, every time that $\zeta>(s)$ changes sing a discontinuous jump by $\pi$ in the phase angle is introduced. Otherwise the phase angle is a smooth function of t. In a previous paper [2], it was noted that as $\rho$ is increased, the phase of the zeta function is smoothed gradually, and the smooth part is directly linked to the quantum scattering phase shift of a one-dimensional inverted harmonic oscillator. In this paper, we first examine the phase of $\zeta(s)$ and show that its derivative with respect to t for a fixed $\rho$ > 1/2 is the Lorentz-smoothed oscillating part of the density of the zeros at $\rho = 1/2$. The latter may also be expressed [3] as a Gutzwiller-like trace formula with primitive orbits whose periods are the logarithms of the prime numbers. By choosing $\rho$ >> 1, the contributions of the larger primes are severely damped. In this limit of large damping, the residual oscillating part of the density of states appears to be of the form of a harmonic oscillator, but with a denominator corresponding to an unstable periodic orbit. In the present paper, we show that this instability in a periodic orbit may be brought about by a very simple model Hamiltonian. In particular, we examine the motion of a particle in a plane, bounded by a harmonic oscillator potential alnong one axis and by a parabolic saddle in the transverse direction. This dynamical model is a caricature of the Riemann zeta function for very large $\rho$. Interestingly enough, the electrostatic potential at the bottleneck of a quantum point contact in a mesoscopic structure [4] has the same shape. In this constriction of the split-gate system, the electron transmission through the saddle may take place in quantised channels, corresponding to the bound states of the one-dimensional harmonic potential. This gives rise to quantised conductance steps that are seen experimentally. The relevance of this will be discussed later. The focus in this paper, however, is not the well-studied transmission problem through this potential, but the semiclassical density of states and its asymptotic connection to the Riemann zeta function. The classical motion perpendicular to the saddle has only one isolated (unstable) periodic orbit and its repetitions. The semiclassical trace formula using the Gutzwiller approach [5] and the Selberg zeta function [6] of this system are obtained analytically. The corresponding bound-state problem of the two- (and higher) dimensional harmonic oscillator has been recently solved by the same method [7]. In our example, the role of the inverted oscillator is to damp out the higher harmonics and improve the convergence of the Selberg zeta function. The quantum mechanical density of states is also obtained analytically. A detailed comparison of the semclassical and quantum results shows that the semiclassical trace formula accurately reproduces the quantum result.

Obviously, this toy model is not the dynamic Hamiltonian that describes the phase of $\zeta(s)$ even outside the strip $\rho$ > 1. But asymptotically, for very large $\rho$, when all but the lowest prime (=2) is dominant, the saddle-like planar potential appears to have relevance. In particular, the curvature of the inverted potential at the saddle is found to be directly proportional to the parameter rho of the zeta function. Recalling that the phase of $\zeta(s)$ on the $\rho$ = 1 line is described by the quantum scattering phase shift of a non-Euclidean surface of constant negative curvature [8], the current work poses the question of whether a single Hamiltonian may describe the phase of $\zeta(s)$ for the entire range $\rho$ >> 1.

Y. Petridis, "Zeros of the Riemann zeta function and central values of L-series of holomorphic cusp forms"

"We examine the stability of the zeros of the Riemann zeta function, which are twice the scattering poles of SL(2,Z), in relation to the central value of the L-series of holomorphic cusp forms of weight 2 for the congruence subgroups $\Gamma_0(q)$, q prime. We work with perturbations in characters varieties of $\Gamma_0(q)$ and study the effects on the spectral and scattering theory of the Laplace operator.

Y. Petridis , "Perturbation of scattering poles for hyperbolic surfaces and central values of L-series", Duke Math. J. 103 no. 1 (2000) 101-130 [DVI file]

R. Delbourgo and A. Ritz, "The effective Lagrangian for low energy photon interactions in any dimension" (preprint 03/95)

"The subject of low energy photon-photon scattering is considered in arbitrary dimensional space-time and the interaction is widened to include scattering events involving an arbitrary number of photons. The effective interaction Lagrangian for these processes in QED has been determined in a manifestly invariant form. This generalisation resolves the structure of the weak-field Euler-Heisenberg Lagrangian and indicates that the component invariant functions have coefficients related, not only to the space-time dimension, but also to the generalised Riemann-Zeta function."

Erik Balslev, "Spectral theory of Laplace operators on hyperbolic Riemann surfaces"

(a course given at the University of Aarhus - in which poles of certain scattering matrices are related to the Riemann zeta zeros)

A. Knauf, "Irregular scattering, number theory, and statistical mechanics" , from Stochasticity and Quantum Chaos (eds. Z. Haba, et. al.) Dordrecht (Kluwer,1995)

Xian-Shun Luo, "Riemann Hypothesis and Levison Theorem" (preprint 12/2008)

[abstract:] "In this paper we will give a simple proof of Riemann Hypothesis, considered to be one of the greatest unsolved problem in mathematics, related to inverse scattering problem and radom matrices. The important relationship between Riemann Hypothesis and random matrices was found by Freeman J. Dyson (1972). Dyson [wrote] a paper [in] 1975 [which] related random matrices and inverse scattering problem. Under this explanation, the famous Riemann Hypothesis is equivalent to Levison theorem of scattering phase-shifts. We will prove this relation."

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