Riemann Zeros Research Articles

Multiparticle Distributions of Ideal Fermi Gas: Analogy to Random Matrices and the Riemann Zeros

The multiparticle density matrices for degenerate, ideal Fermi gas system in any dimension are calculated. The results are expressed as a determinant form, in which a correlation kernel plays a vital role. Interestingly, the correlation structure of one-dimensional Fermi gas system is essentially equivalent to that observed for the eigenvalue distribution of random unitary matrices, and thus to that conjectured for the distribution of the non-trivial zeros of the Riemann zeta function. Implications of the present findings are discussed briefly.

Wigner quantization of some one-dimensional Hamiltonians

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 \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(1|2)$\end{document}osp(1|2). One of them, the Hamiltonian \documentclass[12pt]{minimal}\begin{document}$\hat{H} = \hat{x} \hat{p}$\end{document}Ĥ=x̂p̂, 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, \documentclass[12pt]{minimal}\begin{document}$\hat{H}_f = \hat{p}^2/2$\end{document}Ĥf=p̂2/2, 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 \documentclass[12pt]{minimal}\begin{document}$\mathfrak {osp}(1|2)$\end{document}osp(1|2).

Large– Nc QCD, Harmonic Sums, and the Riemann Zeros

It is shown that in Large-Nc QCD, two--point functions of local operators become Harmonic Sums. We comment on the properties which follow from this fact. This has led us to an aside observation concerning the zeros of the Riemann zeta--function seen from the point of view of Dispersion Relations in Quantum Theory.

A NEW BOUND FOR THE SMALLEST x WITH π(x) > li(x)

We reduce the leading term in Lehman's theorem. This improved estimate allows us to refine the main theorem of Bays and Hudson [2]. Entering 2,000,000 Riemann zeros, we prove that there exists x in the interval [exp (727.951858), exp (727.952178)] for which π(x) - li (x) > 3.2 × 10151. There are at least 10154 successive integers x in this interval for which π(x) > li (x). This interval is strictly a sub-interval of the interval in Bays and Hudson, and is narrower by a factor of about 12.

The Berry–Keating operator on and on compact quantum graphs with general self-adjoint realizations

The Berry–Keating operator (Berry and Keating 1999 SIAM Rev. 41 236) governing the Schrödinger dynamics is discussed in the Hilbert space and on compact quantum graphs. It has been proved that the spectrum of HBK defined on is purely continuous and thus this quantization of HBK cannot yield the hypothetical Hilbert–Polya operator possessing as eigenvalues the nontrivial zeros of the Riemann zeta function. A complete classification of all self-adjoint extensions of HBK acting on compact quantum graphs is given together with the corresponding secular equation in form of a determinant whose zeros determine the discrete spectrum of HBK. In addition, an exact trace formula and the Weyl asymptotics of the eigenvalue counting function are derived. Furthermore, we introduce the ‘squared’ Berry–Keating operator which is a special case of the Black–Scholes operator used in financial theory of option pricing. Again, all self-adjoint extensions, the corresponding secular equation, the trace formula and the Weyl asymptotics are derived for H2BK on compact quantum graphs. While the spectra of both HBK and H2BK on any compact quantum graph are discrete, their Weyl asymptotics demonstrate that neither HBK nor H2BK can yield as eigenvalues the nontrivial Riemann zeros. Some simple examples are worked out in detail.

Quantum mechanical potentials related to the prime numbers and Riemann zeros

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.

Tuck's incompressibility function: statistics for zeta zeros and eigenvalues

For any function D(x) that is real for real x, positivity of Tuck's function Q(x) ≡ D′2(x)/(D′2(x) − D(x)D″(x)) is a condition for the absence of complex zeros close to the real axis. Study of the probability distribution PN(Q), for D(x) with N zeros corresponding to eigenvalues of the Gaussian unitary ensemble (GUE), supports Tuck's observation that large values of Q are very rare for the Riemann zeros. PN(Q) has singularities at Q = 0, Q = 1 and Q = N. The moments (averages of Qm) are much smaller for the GUE than for uncorrelated random (Poisson-distributed) zeros. For the Poisson case, the large-N limit of PN(Q) can be expressed as an integral with infinitely many poles, whose accumulation, requiring regularization with the Lerch transcendent, generates the singularity at Q = 1, while the large-Q decay is determined by the pole closest to the origin. Determining the large-N limit of PN(Q) for the GUE seems difficult.

On the quantum reconstruction of the Riemann zeros

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 ζ(σ − it) for σ > 1. For σ = 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 σ = 1/2, as a Jost function, would imply that the Riemann zeros are point-like spectra embedded in the continuum, resolving in that way the emission/spectral interpretation.

A quantum mechanical model of the Riemann zeros

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 wavefunctions associated with 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 wavefunctions associated with 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 wavefunctions, 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.

Three quantum obsessions

Is there a connection between the Riemann zeros and the quantum physics of classical chaos? Can the relation between spin and statistics be understood within elementary quantum mechanics? How are the phase singularities in classical optics smoothed by quantum effects?

Triple correlation of the Riemann zeros

We use the conjecture of Conrey, Farmer and Zirnbauer for averages of ratios of the Riemann zeta function to calculate all the lower order terms of the triple correlation function of the Riemann zeros. A previous approach was suggested in 1996 by Bogomolny and Keating taking inspiration from semi-classical methods. At that point they did not write out the answer explicitly, so we do that here, illustrating that by our method all the lower order terms down to the constant can be calculated rigourously if one assumes the ratios conjecture of Conrey, Farmer and Zirnbauer. Bogomolny and Keating returned to their previous results simultaneously with this current work, and have written out the full expression. The result presented in this paper agrees precisely with their formula, as well as with our numerical computations, which we include here. We also include an alternate proof of the triple correlation of eigenvalues from random U(N) matrices which follows a nearly identical method to that for the Riemann zeros, but is based on the theorem for averages of ratios of characteristic polynomials.

Correlations of eigenvalues and Riemann zeros

Interest in comparing the statistics of the zeros of the Riemann zeta function with random matrix theory dates back to the 1970s and the work of Montgomery and Dyson. Twelve years ago Rudnick and Sarnak and, independently, Bogomolny and Keating showed that the n-point correlation function of the Riemann zeros, correctly scaled and in the limit of infinite height on the critical line, agrees with the scaling limit of the n-correlation of eigenvalues of random unitary matrices. The former piece of work holds only for a restricted class of test functions, and the latter relies on a heuristic method and the conjectures of Hardy and Littlewood. Neither tells us more than the asymptotic limit for the general n-correlation. In this article we use the ratios conjecture for average values of the Riemann zeta function to produce the lower order terms in a very precise formula for the n-correlation of the Riemann zeros. The same method can be applied rigorously in the random matrix case, yielding a formula which shows identical structure (though with none of the arithmetic details) to the correlations of the Riemann zeros something which cannot be seen from the classical determinantal formula for the random matrix correlation functions.

Formula omitted] with interaction and the Riemann zeros

Starting from a quantized version of the classical Hamiltonian H = x p , 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 = x p 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.

Fractal fits to Riemann zeros

Wu and Sprung (Phys. Rev. E, 48, 2595 (1993)) reproduced the first 500 nontrivial Riemann zeros, using a one-dimensional local potential model. They concluded — as did van Zyl and Hutchinson (Phys. Rev. E, 67, 066211 (2003)) — that the potential possesses a fractal structure of dimension d = 3/2. We model the nonsmooth fluctuating part of the potential by the alternating-sign sine series fractal of Berry and Lewis A(x,γ). Setting d = 3/2, we estimate the frequency parameter (γ), plus an overall scaling parameter (σ) that we introduce. We search for that pair of parameters (γ,σ) that minimizes the least-squares fit Sn(γ,σ) of the lowest n eigenvalues — obtained by solving the one-dimensional stationary (nonfractal) Schrodinger equation with the trial potential (smooth plus nonsmooth parts) — to the lowest n Riemann zeros for n = 25. For the additional cases, we study, n = 50 and 75, we simply set σ = 1. The fits obtained are compared to those found by using just the smooth part of the Wu–Sprung potential without any fractal supplementation. Some limited improvement — 5.7261 versus 6.392 07 (n = 25), 11.2672 versus 11.7002 (n = 50), and 16.3119 versus 16.6809 (n = 75) — is found in our (nonoptimized, computationally bound) search procedures. The improvements are relatively strong in the vicinities of γ = 3 and (its square) 9. Further, we extend the Wu-Sprung semiclassical framework to include higher order corrections from the Riemann–von Mangoldt formula (beyond the leading, dominant term) into the smooth potential. PACS Nos.: 02.10.De, 03.65.Sq, 05.45.Df, 05.45.Mt

The Riemann zeros and the cyclic renormalization group

We propose a consistent quantization of the Berry–Keating Hamiltonianxp, which is currently discussed in connection with the non-trivial zeros of the Riemannzeta function. The smooth part of the Riemann counting formula for the zeros isreproduced exactly. The zeros appear, not as eigenstates, but as missing states in thespectrum, in agreement with Connes adelic approach to the Riemann hypothesis.The model is exactly solvable and renormalizable, with a cyclic renormalizationgroup. These results are obtained by mapping the Berry–Keating model into theRussian doll model of superconductivity. Finally, we propose a generalization ofthese models in an attempt to explain the oscillatory part of Riemann’s formula.

Power spectrum of the fluctuation of Chebyshev's prime counting function

The one-sided power spectrum of the fluctuation of Chebyshev's weighted prime counting function is numerically estimated based on samples of the fluctuating function of different sizes. The power spectrum is also estimated analytically for large frequency based on Riemann hypothesis and the exact formula for the fluctuating function in terms of all the non-trivial Riemann zeroes. Our analytical estimate is consistent with our numerical estimate of a 1 / f 2 power spectrum.

Two complete and minimal systems associated with the zeros of the Riemann zeta function

Two complete and minimal systems associated with the zeros of the Riemann zeta function

Riemann zeros, prime numbers, and fractal potentials

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, Delta(3), of the energy levels.

Inverse scattering, the coupling constant spectrum, and the Riemann hypothesis

It is well known that the s-wave Jost function for a potential, λV, is an entire function of λ with an infinite number of zeros extending to infinity. For a repulsive V, and at zero energy, these zeros of the ‘coupling constant’, λ, will all be real and negative, λn(0)<0. By rescaling λ, such that λn<−1/4, and changing variables to s, with λ=s(s−1), it follows that as a function of s the Jost function has only zeros on the line sn=1/2+iγn. Thus, finding a repulsive V whose coupling constant spectrum coincides with the Riemann zeros will establish the Riemann hypothesis, but this will be a very difficult and unguided search.

Spectral Spacing Correlations for Chaotic and Disordered Systems

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.