Riemann Zeros Research Articles

Hamiltonian for the Hilbert–Pólya conjecture

We introduce a Hamiltonian to address the Hilbert–Pólya conjecture. The eigenfunctions of the introduced Hamiltonian, subject to the Dirichlet boundary conditions on the positive half-line, vanish at the origin by the nontrivial zeros of the Riemann zeta function. Consequently, the eigenvalues are determined by these nontrivial Riemann zeros. If the Riemann hypothesis (RH) is true, the eigenvalues become real and represent the imaginary parts of the nontrivial zeros. Conversely, if the Hamiltonian is self-adjoint, or more generally, admits only real eigenvalues, then the RH follows. In our attempt to demonstrate the latter, we establish the existence of a similarity transformation of the introduced Hamiltonian that is self-adjoint on the domain specified by an appropriate boundary condition, which is satisfied by the eigenfunctions through the vanishing of the Riemann zeta function. Our result can be extended to a broader class of functions whose zeros lie on the critical line.

Supersymmetric quantum mechanical system for locating the Riemann zeros

Supersymmetric quantum mechanical system for locating the Riemann zeros

The boundary between the Prime Gaps and the Interchangeable be-tween a Quantum Mechanics System & a Classical Chaos One

With reference to my last paper, this author found that there are always some discrete quantum constants like the Planck’s one laying between the real and the imaginary parts of the Riemann function (I.e when we fix or group the Riemann Zeros one by one and make the real part to be varying.) or the quantum mechan-ics system. Hence, the Riemann Zeta Zeros give us the discrete quantum con-stants together with the given discrete quantum constants may give you the wanted Riemann Zeta Zeros. This phenomenon may be explained by Quantum Riemann Zeta Epiphancy. In the present paper, this author tries to fix or group the real part of the Riemann Zeros and make the imaginary part of the Riemann Zeros to be varying from one to the tenth one or resulting a chaos like graph. Thus, it seems that we may interchange a quantum mechanics system with the classical chaos one by just varying one variable and fix another one (such as varying real part of the Riemann function’s zeors from 0.1 to 1 and fix the imaginary part of the Rie-mann zeros one group after one another). Certainly, the vice versa for the fix of the real part (one group after another) with different Riemann Zeros may also help. In other words, the quantum mechanics system is just the conjugate pair (varying x and fixed y variables) of the classical chaos one or the vice versa is al-so true (fixed x and varying y variables) & put back into the Zeta functions etc. Hence, it is a method (or transformation) for us so that we may easily to make a interchange between these two systems by only adjusting or selecting which vari-ables to be varying or fixed. Then we may turn the classical chaos system into a quantum mechanics one or the vice versa. This implies that it may be possible to apply controllable quantum systems for the investigation of our natural aspects. Finally, this author have also shown that there is a boundary laying be-tween the prime gap through the elementary calculation and mathematical deriva-tion without using the computer software etc. It is thus a contradiction to the fact that there is no bound for the prime gap. This tells us that there is a need to have a shift from zero to 0.5 for the real part of the Riemann Zeta Zeros as what this author has mentioned in my last paper etc.

Riemann Hypothesis and Random Walks: The Zeta Case

In previous work, it was shown that if certain series based on sums over primes of non-principal Dirichlet characters have a conjectured random walk behavior, then the Euler product formula for its L-function is valid to the right of the critical line ℜ(s)>12, and the Riemann hypothesis for this class of L-functions follows. Building on this work, here we propose how to extend this line of reasoning to the Riemann zeta function and other principal Dirichlet L-functions. We apply these results to the study of the argument of the zeta function. In another application, we define and study a one-point correlation function of the Riemann zeros, which leads to the construction of a probabilistic model for them. Based on these results we describe a new algorithm for computing very high Riemann zeros, and we calculate the googol-th zero, namely 10100-th zero to over 100 digits, far beyond what is currently known. Of course, use is made of the symmetry of the zeta function about the critical line.

Riemann zeros from Floquet engineering a trapped-ion qubit

The non-trivial zeros of the Riemann zeta function are central objects in number theory. In particular, they enable one to reproduce the prime numbers. They have also attracted the attention of physicists working in random matrix theory and quantum chaos for decades. Here we present an experimental observation of the lowest non-trivial Riemann zeros by using a trapped-ion qubit in a Paul trap, periodically driven with microwave fields. The waveform of the driving is engineered such that the dynamics of the ion is frozen when the driving parameters coincide with a zero of the real component of the zeta function. Scanning over the driving amplitude thus enables the locations of the Riemann zeros to be measured experimentally to a high degree of accuracy, providing a physical embodiment of these fascinating mathematical objects in the quantum realm.

Prime zeta function statistics and Riemann zero-difference repulsion

We present a derivation of the numerical phenomenon in which differences between the Riemann zeta function’s nontrivial zeros tend to avoid being equal to the imaginary parts of the zeros themselves, a property called statistical ‘repulsion’ between the zeros and their differences. Our derivation relies on the statistical properties of the prime zeta function, whose singularity structure specifies the positions of the Riemann zeros. We show that the prime zeta function on the critical line is asymptotically normally distributed with a covariance function that is closely approximated by the logarithm of the Riemann zeta function’s magnitude on the one-line. This creates notable negative covariance at separations approximately equal to the imaginary parts of the Riemann zeros. This covariance function and the singularity structure of the prime zeta function combine to create a conditional statistical bias at the locations of the Riemann zeros that predicts the zero-difference repulsion effect. Our method readily generalizes to describe similar effects in the zeros of related L-functions.

Identifying the Riemann zeros by periodically driving a single qubit

The Riemann hypothesis, one of the most important open problems in pure mathematics, implies the most profound secret of prime numbers. One of the most interesting approaches to solve this hypothesis is to connect the problem with the spectrum of the physical Hamiltonian of a quantum system. However, none of the proposed quantum Hamiltonians have been experimentally feasible.Here, we report the first experiment to identify the first non-trivial zeros of the Riemann zeta function and the first two zeros of P\'olya's fake zeta function, using a novel Floquet method, through properly designed periodically driving functions. According to this method, the zeros of these functions are characterized by the occurrence of crossings of quasi-energies when the dynamics of the system are frozen. The experimentally obtained zeros are in excellent agreement with their exact values. Our study provides the first experimental realization of the Riemann zeros, which may provide new insights into this fundamental mathematical problem.

Twin prime correlations from the pair correlation of Riemann zeros

We establish, via a heuristic Fourier inversion calculation, that the Hardy–Littlewood twin prime conjecture is equivalent to an asymptotic formula for the two-point correlation function of Riemann zeros at a height E on the critical line. Previously it was known that the Hardy–Littlewood conjecture implies the pair correlation formula, and we show that the reverse implication also holds. An averaged form of the Hardy–Littlewood conjecture is obtained by inverting the limit of the two-point correlation function and the precise form of the conjecture is found by including asymptotically lower order terms in the two-point correlation function formula.

The Riemann Zeros as Spectrum and the Riemann Hypothesis

We present a spectral realization of the Riemann zeros based on the propagation of a massless Dirac fermion in a region of Rindler spacetime and under the action of delta function potentials localized on the square free integers. The corresponding Hamiltonian admits a self-adjoint extension that is tuned to the phase of the zeta function, on the critical line, in order to obtain the Riemann zeros as bound states. The model suggests a proof of the Riemann hypothesis in the limit where the potentials vanish. Finally, we propose an interferometer that may yield an experimental observation of the Riemann zeros.

Supersymmetry and the Riemann zeros on the critical line

We propose a new way of studying the Riemann zeros on the critical line using ideas from supersymmetry. Namely, we construct a supersymmetric quantum mechanical model whose energy eigenvalues correspond to the Riemann zeta function in the strip 0<Res<1 (in the complex parameter space) and show that the zeros on the critical line arise naturally from the vanishing ground state energy condition in this model.

Quantization of Non-standard Hamiltonians and the Riemann Zeros

Relations between number theory, quantum mechanics and statistical mechanics are of interest to mathematicians and physicists since it was suggested that the zeros of the Riemann zeta function might be related to the spectrum of a self-adjoint quantum mechanical operator related to a one-dimensional Hamiltonian $$ H = xp $$ known as Berry–Keating–Connes Hamiltonian. However, this type of Hamiltonian is integrable and the classical trajectories of particles are not closed leading to a continuum spectrum. Recently, Sierra and Rodriguez-Laguna conjectured that the Hamiltonian $$ H = x(p \,+\, {\xi \mathord{\left/ {\vphantom {\xi p}} \right. \kern-0pt} p}) $$ where $$ \xi $$ is a coupling constant with dimensions of momentum square is characterized by a quantum spectrum which coincides with the average Riemann zeros and contains closed periodic orbits. In this paper, we show first that the Sierra–Rodriguez-Laguna Hamiltonian may be obtained by means of non-standard singular Lagrangians and besides the Hamiltonians $$ H = x(p + {\xi \mathord{\left/ {\vphantom {\xi p}} \right. \kern-0pt} p}) $$ and $$ H(x,p) = px $$ are not the only semiclassical Hamiltonians connected to the average Riemann zeros. We show the presence of a new Hamiltonian where its quantization revealed a number of interesting properties, in particular, the sign of a trace of the Riemann zeros.

On Prime Numbers and The Riemann Zeros

The current research regarding the Riemann zeros suggests the existence of a non-trivial algebraic/analytic structure on the set of Riemann zeros. The duality between primes and Riemann zeta function zeros suggests some new goals and aspects to be studied: {\em adelic duality} and the {\em POSet of prime numbers}. The article presents computational evidence of the structure of the imaginary parts $t$ of the non-trivial zeros of the Riemann zeta function $\rho=1/2+it$, called in this article the {\em Riemann Spectrum}, using the study of their distribution. The novelty represents in considering the associated characters $p^{it}$, towards an algebraic point of view, than rather in the sense of Analytic Number Theory. This structure is tentatively interpreted in terms of adelic characters, and the duality of the rationals. Second, the POSet structure of prime numbers studied, is tentatively mirrored via duality in the Riemann spectrum. A direct study of the convergence of their Fourier series, along Pratt trees, is proposed. Further considerations, relating the Riemann Spectrum, adelic characters and distributions, in terms of Hecke idelic characters, local zeta integrals (Mellin transform) and $\omega$-eigen-distributions, are explored following.

On the value distribution of two Dirichlet $\boldsymbol L$-functions

Let $\rho$ denote the non-trivial zeros of the Riemann zeta function. We study the relative value distribution of $L(\rho+\sigma,\chi_{1})$ and $L(\rho+\sigma,\chi_{2})$, where $\sigma\in[0,1/2)$ is fixed and $\chi_{1}$, $\chi_{2}$ are two fixed Dirichlet characters to distinct prime moduli. For $\sigma>0$ we prove that a positive proportion of these pairs of values are linearly independent over $\mathbb{R}$, which implies that the arguments of the values are different. For $\sigma=0$ we show that, up to height $T$, the values are different for $cT$ of the Riemann zeros for some positive constant $c$.

Finite Size Effects for Spacing Distributions in Random Matrix Theory: Circular Ensembles and Riemann Zeros

According to Dyson's threefold way, from the viewpoint of global time reversal symmetry, there are three circular ensembles of unitary random matrices relevant to the study of chaotic spectra in quantum mechanics. These are the circular orthogonal, unitary, and symplectic ensembles, denoted COE, CUE, and CSE, respectively. For each of these three ensembles and their thinned versions, whereby each eigenvalue is deleted independently with probability , we take up the problem of calculating the first two terms in the scaled large N expansion of the spacing distributions. It is well known that the leading term admits a characterization in terms of both Fredholm determinants and Painlevé transcendents. We show that modifications of these characterizations also remain valid for the next to leading term, and that they provide schemes for high precision numerical computations. In the case of the CUE, there is an application to the analysis of Odlyzko's data set for the Riemann zeros, and in that case, some further statistics are similarly analyzed.

Notes on $\log (\zeta (s))^\prime \prime $

Motivated by the connection to the pair correlation of the Riemann zeros, we investigate the second derivative of the logarithm of the Riemann $\zeta $ function, in particular, the zeros of this function. Theorem~1.2 gives a zero-free region. Theorem~1.4 gives an asymptotic estimate for the number of nontrivial zeros to height $T$. Theorem~1.7 is a zero density estimate.

Can one hear the Riemann zeros in black hole ringing?

We elaborate on an entry of the AdS/CFT dictionary relating functional determinants: the determinant of the one-loop contribution to the effective gravitational action by bulk scalars in an asymptotically locally AdS background X, and the determinant of the two-point function of the dual operator (a.k.a. scattering matrix) at the conformal boundary M. The formula originates from AdS/CFT heuristics that map a quantum contribution in the bulk gravitational partition function to a subleading large-N contribution in the boundary CFT partition function:The formula applies to quotients of AdS as well [1]. In the particular case of the BTZ black hole, a closed expression can be worked out in terms of an associated Patterson-Selberg zeta function ZBTZ(λ) [2]. The determinants can then be thought of as regularized products of either zeta zeros, scattering resonances or quasinormal frequencies [3]. In this sense, one could say that the zeros of ZBTZ(λ) can be heard in the spectrum of quasinormal modes of the BTZ black hole. The question we want to pose is whether a similar situation might exist for the celebrated zeros of the Riemann zeta function.

On Riemann zeroes, lognormal multiplicative chaos, and Selberg integral

Rescaled Mellin-type transforms of the exponential functional of the Bourgade–Kuan–Rodgers statistic of Riemann zeroes are conjecturally related to the distribution of the total mass of the limit lognormal stochastic measure of Mandelbrot–Bacry–Muzy. The conjecture implies that a non-trivial, log-infinitely divisible probability distribution is associated with Riemann zeroes. For application, integral moments, covariance structure, multiscaling spectrum, and asymptotics associated with the exponential functional are computed in closed form using the known meromorphic extension of the Selberg integral.

Whittaker vector, Wheeler-DeWitt equation, and the gravity dual of conformal quantum mechanics

We study the energy representation of conformal quantum mechanics as the Whittaker vector without specifying classical Lagrangian. We show that a generating function of expectation values among two excited states of the dilatation operator in conformal quantum mechanics is a solution to the Wheeler-DeWitt equation and it corresponds to the AdS2 partition function evaluated as the minisuperspace wave function in Liouville field theory. We also show that the dilatation expectation values in conformal quantum mechanics lead to the asymptotic smoothed counting function of the Riemann zeros.

A procedure on the first integrals of second-order nonlinear ordinary differential equations

In this article, we demonstrate the applicability of the integrating factor method to path equation describing minimum drag work, and a special Hamiltonian equation corresponding Riemann zeros for obtaining the first integrals. The effectiveness and powerfullness of this method is verified by applying it for two selected second-order nonlinear ordinary differential equations (NLODEs). As a result integrating factors and first integrals for them are succesfully established. The obtained results show that the integrating factor approach can also be applied to other NLODEs.

Finite-size corrections in random matrix theory and Odlyzko’s dataset for the Riemann zeros

Odlyzko has computed a dataset listing more than 109successive Riemann zeros, starting from a zero number to beyond 1023. This dataset relates to random matrix theory as, according to the Montgomery–Odlyzko law, the statistical properties of the large Riemann zeros agree with the statistical properties of the eigenvalues of large random Hermitian matrices. Moreover, Keating and Snaith, and then Bogomolny and co-workers, have usedN×Nrandom unitary matrices to analyse deviations from this law. We contribute to this line of study in two ways. First, we point out that a natural process to apply to the dataset is to minimize it by deleting each member independently with some specified probability, and we proceed to compute empirical two-point correlation functions and nearest neighbour spacings in this setting. Second, we show how to characterize the order 1/N2correction term to the spacing distribution for random unitary matrices in terms of a second-order differential equation with coefficients that are Painlevé transcendents, and where the thinning parameter appears only in the boundary condition. This equation can be solved numerically using a power series method. In comparison to the Riemann zero data accurate agreement is exhibited.