Riemann Zeros Research Articles

Riemann zeros in radiation patterns: II. Fourier transforms of zeta

This extends a previous study (2012 J. Phys. A: Math. Theor. 45 302001) of two initial waveforms whose far-field radiation patterns possess sidelobes separated by the Riemann zeros. The analysis suffered from the disadvantage that the sidelobes were very weak, making it difficult to detect the zeros between them. To overcome this, new Fourier pairs are derived, whose sidelobes are not weak. These are transforms of the zeta function on the critical line, modulated by functions with no zeros on the line.

Finding zeros of the Riemann zeta function by periodic driving of cold atoms

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 P\'olya-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.

One-loop effective action and the Riemann zeros

We present a remarkable connection between the asymptotic behavior of the Riemann zeros and one-loop effective action in Euclidean scalar field theory. We show that in a two-dimensional space, the asymptotic behavior of the Fourier transform of two-point correlation functions fits the asymptotic distribution of the nontrivial zeros of the Riemann zeta function. We work out an explicit example, namely the nonlinear sigma model in the leading order in 1/N expansion.

Unnormalized differences between zeros ofL-functions

AbstractWe study a subtle inequity in the distribution of unnormalized differences between imaginary parts of zeros of the Riemann zeta function, which was observed by a number of authors. We establish a precise measure which explains the phenomenon, that the location of each Riemann zero is encoded in the distribution of large Riemann zeros. We also extend these results to zeros of more general$L$-functions. In particular, we show how the rank of an elliptic curve over$\mathbb{Q}$is encoded in the sequences of zeros ofother$L$-functions, not only the one associated to the curve.

The Riemann zeros as energy levels of a Dirac fermion in a potential built from the prime numbers in Rindler spacetime

We construct a Hamiltonian HR whose discrete spectrum contains, in a certain limit, the Riemann zeros. HR is derived from the action of a massless Dirac fermion living in a domain of Rindler spacetime, in 1 + 1 dimensions, which 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 that are associated with the prime numbers p, whose periods, as measured by the observerʼs clock, are . Acting on the chiral components of the fermion , HR becomes the Berry–Keating Hamiltonian , where x is identified with the Rindler spatial coordinate and 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 ϑ, the spectrum is a continuum where the Riemann zeros are missing, as in the adelic Connes model. However, for some values of ϑ, related to the phase of the zeta function, the Riemann zeros appear as discrete eigenvalues that are immersed in the continuum. We generalize this result to the zeros of Dirichlet L-functions, which are 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 the Random Matrix Theory.

Riemann zeta zeros and zero-point energy

The sequence of nontrivial zeros of the Riemann zeta function is zeta regularizable. Therefore, systems with countably infinite number of degrees of freedom described by self-adjoint operators whose spectra is given by this sequence admit a functional integral formulation. We discuss the consequences of the existence of such self-adjoint operators in field theory framework. We assume that they act on a massive scalar field coupled to a background field in a (d+1)-dimensional flat space–time where the scalar field is confined to the interval [0, a] in one of its dimensions and there are no restrictions in the other dimensions. The renormalized zero-point energy of this system is presented using techniques of dimensional and analytic regularization. In even-dimensional space–time, the series that defines the regularized vacuum energy is finite. For the odd-dimensional case, to obtain a finite vacuum energy per unit area, we are forced to introduce mass counterterms. A Riemann mass appears, which is the correction to the mass of the field generated by the nontrivial zeros of the Riemann zeta function.

Spectral realization of the Riemann zeros by quantizing H = w(x)(p + ℓ2p/p): the Lie-Noether symmetry approach

If tn are the heights of the Riemann zeros 1/2 + itn, an old idea, attributed to Hilbert and Polya [6], stated that the Riemann hypothesis would be proved if the tn could be shown to be eigenvalues of a self-adjoint operator. In 1986 Berry [1] conjectured that tn could instead be the eigenvalues of a deterministic quantum system with a chaotic classical counterpart and in 1999 Berry and Keating [3] proposed the Hamiltonian H = xp, with x and p the position and momentum of a one-dimensional particle, respectively. This was proven not to be the correct Hamiltonian since it yields a continuum spectrum [23] and therefore a more general Hamiltonian H = w(x)(p + ℓ2p/p) was proposed [25], [4], [24] and different expressions of the function w(x) were considered [25], [24], [16] although none of them yielding exactly tn. We show that the quantization by means of Lie and Noether symmetries [18], [19], [20], [7] of the Lagrangian equation corresponding to the Hamiltonian H yields straightforwardly the Schrödinger equation and clearly explains why either the continuum or the discrete spectrum is obtained. Therefore we infer that suitable Lie and Noether symmetries of the classical Lagrangian corresponding to H should be searched in order to alleviate one of Berry's quantum obsessions [2].

Quantum Graphs Whose Spectra Mimic the Zeros of the Riemann Zeta Function

One of the most famous problems in mathematics is the Riemann hypothesis: that the nontrivial 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.

AN ELECTROSTATIC DEPICTION OF THE VALIDITY OF THE RIEMANN HYPOTHESIS AND A FORMULA FOR THE NTH ZERO AT LARGE N

We construct a vector field E from the real and imaginary parts of an entire function ξ(z) which arises in the quantum statistical mechanics of relativistic gases when the spatial dimension d is analytically continued into the complex z plane. This function is built from the Γ and Riemann ζ functions and is known to satisfy the functional identity ξ(z) = ξ(1-z). E satisfies the conditions for a static electric field. The structure of E in the critical strip is determined by its behavior near the Riemann zeros on the critical line ℜ(z) = 1/2, where each zero can be assigned a ⊕ or ⊖ parity, or vorticity, of a related pseudo-magnetic field. Using these properties, we show that a hypothetical Riemann zero that is off the critical line leads to a frustration of this "electric" field. We formulate this frustration more precisely in terms of the potential Φ satisfying E = -∇Φ and construct Φ explicitly. The main outcome of our analysis is a formula for the nth zero on the critical line for large n expressed as the solution of a simple transcendental equation. Riemann's counting formula for the number of zeros on the entire critical strip can be derived from this formula. Our result is much stronger than Riemann's counting formula, since it provides an estimate of the nth zero along the critical line. This provides a simple way to estimate very high zeros to very good accuracy, and we estimate the 10106-th one.

RIEMANN ZEROS AND THE INVERSE PHASE PROBLEM

Finding a universal method of crystal structure solution and proving the Riemann hypothesis are two outstanding challenges in apparently unrelated fields. For centrosymmetric crystals however, a connection arises as the result of a statistical approach to the inverse phase problem. It is shown that parameters of the phase distribution are related to the non-trivial Riemann zeros by a Mellin transform.

A method for calculating spectral statistics based on random-matrix universality with an application to the three-point correlations of the Riemann zeros

We illustrate a general method for calculating spectral statistics that combines the universal (random matrix theory limit) and the non-universal (trace-formula-related) contributions by giving a heuristic derivation of the three-point correlation function for the zeros of the Riemann zeta function. The main idea is to construct a generalized Hermitian random matrix ensemble whose mean eigenvalue density coincides with a large but finite portion of the actual density of the spectrum or the Riemann zeros. Averaging the random matrix result over remaining oscillatory terms related, in the case of the zeta function, to small primes leads to a formula for the three-point correlation function that is in agreement with results from other heuristic methods. This provides support for these different methods. The advantage of the approach we set out here is that it incorporates the determinantal structure of the random matrix limit.

A Dirac-type variant of the xp-model and the Riemann zeros

We propose a Dirac-type modification of the xp-model to a model on a semi-infinite cylinder. This model is inspired by recent work of Sierra et al. on the xp-model on the half-line. Our model realizes the Berry-Keating conjecture on the Riemann zeros. We indicate the connection of our model to that of gapped graphene with a supercritical Coulomb charge, which might provide a physical system for the study of the zeros of the Riemann Zeta function.

Hearing the music of the primes: auditory complementarity and the siren song of zeta

A counting function for the primes can be rendered as a sound signal whose harmonies, spanning the gamut of musical notes, are the Riemann zeros. But the individual primes cannot be discriminated as singularities in this ‘music’, because the intervals between them are too short. Conversely, if the prime singularities are detected as a series of clicks, the Riemann zeros correspond to frequencies too low to be heard. The sound generated by the Riemann zeta function itself is very different: a rising siren howl, which can be understood in detail from the Riemann–Siegel formula.

Riemann zeros in radiation patterns

Propagation into the far field changes initial waves into their Fourier transforms. This implies that the Riemann zeros could be observed experimentally in the radiation pattern generated by an initial wave whose Fourier transform is proportional to the Riemann zeta function on the critical line. Two such waves are examined, generating the Riemann Ξ(t) function (pattern 1) and the function ζ(1/2 + it)/(1/2 + it) (pattern 2). For pattern 1, the radiation side lobes are probably too weak to allow detection of the zeros, but for pattern 2 the lobes are stronger, suggesting a feasible experiment.

The pair correlation of homotetic images of zeros of the Riemann zeta-function

We investigate the pair correlation of zeros of the Riemann zeta-function, ζ(s), and zeros of the function ζ(k(s−1/2)+1/2), for any fixed positive integer k.

General covariant xp models and the Riemann zeros

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 HI = x(p + 1/p) and HII = (x + 1/x)(p + 1/p), which have been recently discussed in connection with the nontrivial 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 HI corresponds to a flat spacetime, whereas its spectrum approaches the Riemann zeros on average. The latter property also holds for the model HII, 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.

FLUCTUATIONS OF THE RIEMANN STAIRCASE

The one-sided power spectrum of the fluctuations of the Riemann staircase is calculated numerically and shown to be Lorentzian, like the one-sided power spectrum of the fluctuations of the spectral staircase for classically chaotic systems. This result provides further evidence in support of Berry's conjecture that the En's of the enigmatic Riemann zeroes 1/2 + iEn are real because they are the quantized energies of a system which is classically chaotic.

Nonclassical Degrees of Freedom in the Riemann Hamiltonian

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 the Altland-Zirnbauer universality classC. 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. This resolves a previously mysterious sign problem with the oscillatory contributions to the density of the Riemann zeros.

A compact hamiltonian with the same asymptotic mean spectral density as the Riemann zeros

For the classical hamiltonian (x + 1/x)(p + 1/p), with position x and conjugate momentum p, all orbits are bounded. After a symmetrization, the corresponding quantum integral equation possesses a family of self-adjoint extensions: compact operators on the entire positive x axis, labelled by an angle α specifying the boundary condition at the origin, with a discrete spectrum of real energies E. On the cylinder {−∞ < E < ∞, 0 ≤ α < 2π}, there is a single eigencurve in the form of a helix winding clockwise. The rise between successive windings gets sharper as the scaled Planck's constant decreases. This behaviour can be understood semiclassically. The first two terms of the asymptotic eigenvalue density are the same as those for the density of heights of the Riemann zeros.

H=xpModel Revisited and the Riemann Zeros

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, on average, as the nontrivial zeros of the Riemann zeta function. However, the classical trajectories are not closed, rendering the model incomplete. In this Letter, 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 Polya's fake zeta function and suggest an experimental realization in terms of the Landau model.