# Get e-book The Riemann Hypothesis for Function Fields: Frobenius Flow and Shift Operators

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. Sierra and J. 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. This result is generalized to Dirichlet L-functions using different self-adjoint extensions of H. We show that all these models are covariant under general coordinate transformations. General covariance is maintained by quantization and we find that the spectra are closely related to the geometry of the associated spacetimes. These results suggest the existence of a Hamiltonian whose underlying spacetime encodes the prime numbers, and whose spectrum provides the Riemann zeros.

The action contains a sum of delta function potentials that can be viewed as partially reflecting moving mirrors. 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.

## Riemann zeta functions and linear operators

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. Creffield and G. 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. 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.

So far no such operator was found. 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.

In other words we show that the generating functional of connected Schwinger functions of the associated quantum field theories cannot be constructed. Regniers and J. Such a classification has already been made by J. Hughes, but we reexamine this classification using elementary arguments. Bump, Kwok-Kwong Choi, P. Kurlberg, and J. A subscription to Mathematische Zeitschrift is required if you wish to download this. Aneva, "Symmetry in phase space of a chaotic system" , from AIP conference proceedings Disordered and Complex Systems Springer, — [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.

Schumayer, B. Hutchinson, "Quantum mechanical potentials related to the prime numbers and Riemann zeros" , Phys. E 78 [abstract:] "Prime numbers are the building blocks of our arithmetic; however, their distribution still poses fundamental questions.

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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. Our results offer hope for further analytical progress. Sakhr, R. Bhaduri and B. E 68 [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. Hutchinson, "Riemann zeros, prime numbers, and fractal potentials" , Phys.

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E 67 [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. 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.

Crehan, "Chaotic spectra of classically integrable systems" , Journal of Physics A 28 "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.

They were " withdrawn by arXiv administrators because of fraudulently claimed institutional affiliation and status ". Here , Moreta claims that this was due to a misunderstanding.

It is shown how this can be put into an integro-differential form of a type recently considered by Sierra. Burnol , "A lower bound in an approximation problem involving the zeros of the Riemann zeta function" , Advances in Mathematics [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. Joffily, "A model for the quantum vacuum" , Nucl. A c—c [abstract:] "Following our recent works [S.

Aldrovandi, et al. We then present a description of the vacuum structure as being a dynamical system described by "virtual resonances", completely independent of the second quantization. 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. Bohigas, P. Leboeuf, M. 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.

Leboeuf , and M. Sanchez, "Spectral spacing correlations for chaotic and disordered systems" , Found. They can be viewed as a discretized two point correlation function. Their behavior results from two different contributions.

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## number theory - Books about the Riemann Hypothesis - Mathematics Stack Exchange

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.

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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. Leboeuf, A. Monastra and O. The imaginary part of the zeros are interpreted as mean-field single-particle energies, and one fills them up to a Fermi energy E F.

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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. Leboeuf and A. Monastra, "Quantum thermodynamic fluctuations of a chaotic Fermi-gas model" , Nucl. A [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. Timberlake and J. Tucker, "Is there quantum chaos in the prime numbers? 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.