Physics of the Riemann Hypothesis

Abstract

Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a particular number theoretical function, the Riemann zeta function and examine its influence in the realm of physics and also how physics may be suggestive for the resolution of one of mathematics' most famous unconfirmed conjectures, the Riemann Hypothesis. Does physics hold an essential key to the solution for this more than hundred-year-old problem? In this work we examine numerous models from different branches of physics, from classical mechanics to statistical physics, where this function plays an integral role. We also see how this function is related to quantum chaos and how its pole-structure encodes when particles can undergo Bose-Einstein condensation at low temperature. Throughout these examinations we highlight how physics can perhaps shed light on the Riemann Hypothesis. Naturally, our aim could not be to be comprehensive, rather we focus on the major models and aim to give an informed starting point for the interested Reader.

27 pages, 9 figures

References in corpus (37)

• Ray and wave chaos in asymmetric resonant optical cavities
• Observation of Landau levels of Dirac fermions in graphite
• Random Matrices and Chaos in Nuclear Physics
• Phase Transition in the Number Partitioning Problem
• Landau Levels and Quantum Hall Effect in Graphene Superlattices
• Spectral Statistics: From Disordered to Chaotic Systems
• Random Matrices and Chaos in Nuclear Spectra
• On the rate of quantum ergodicity in Euclidean billiards
• Extreme Value Statistics and Traveling Fronts: An Application to Computer Science
• Measuring Scars of Periodic Orbits
• Open circular billiards and the Riemann hypothesis
• Comments on the Sign and Other Aspects of Semiclassical Casimir Energies
• Fermion-boson transmutation and comparison of statistical ensembles in one dimension
• Maxwell's Demon at work: Two types of Bose condensate fluctuations in power-law traps
• Coupled Microwave Billiards as a Model for Symmetry Breaking
• The Phase of the Riemann Zeta Function and the Inverted Harmonic Oscillator
• The Quantum Mellin transform
• H = x p with interaction and the Riemann zeros
• Quantum mechanical potentials related to the prime numbers and Riemann zeros
• Overcoming the wall in the semiclassical baker's map
• Riemann zeros, prime numbers and fractal potentials
• Duality, Partial Supersymmetry, and Arithmetic Number Theory
• Quantum hamiltonians and prime numbers
• The Quantum Mechanical Potential for the Prime Numbers
• Asymptotics of the number partitioning distribution
• A quantum mechanical model of the Riemann zeros
• Factorising numbers with a Bose-Einstein condensate
• Zeta Function Zeros, Powers of Primes, and Quantum Chaos
• Phase Transition in Multiprocessor Scheduling
• Interacting Bose and Fermi gases in low dimensions and the Riemann hypothesis
• Symmetry of the Riemann Operator
• Chaotic Spectra of Classically Integrable Systems
• Comment on "Bose-Einstein condensation in low-dimensional traps"
• Quantum correlations from Brownian diffusion of chaotic level-spacings
• Quantum statistical mechanics of gases in terms of dynamical filling fractions and scattering amplitudes
• Entropy-based analysis of the number partitioning problem
• Number Fluctuation and the Fundamental Theorem of Arithmetic

Cited by in corpus (62)

• Synthetic Unruh effect in cold atoms
• The H=xp model revisited and the Riemann zeros
• Will a physicist prove the Riemann Hypothesis?
• New horizons in multidimensional diffusion: The Lorentz gas and the Riemann Hypothesis
• The Riemann zeros as energy levels of a Dirac fermion in a potential built from the prime numbers in Rindler spacetime
• The Riemann zeros as spectrum and the Riemann hypothesis
• Quantum theory of bright matter wave solitons in harmonic confinement
• Transcendental equations satisfied by the individual zeros of Riemann $\zeta$, Dirichlet and modular $L$-functions
• General covariant xp models and the Riemann zeros
• Grand-canonical condensate fluctuations in weakly interacting Bose-Einstein condensates of light
• Finding zeros of the Riemann zeta function by periodic driving of cold atoms
• Riemannian Quantum Circuit
• The complex architecture of primes and natural numbers
• Riemann zeta zeros and prime number spectra in quantum field theory
• Majorana quanta, string scattering, curved spacetimes and the Riemann Hypothesis
• Identifying the Riemann zeros by periodically driving a single qubit
• Hilbert-Pólya Conjecture, Zeta-Functions and Bosonic Quantum Field Theories
• Statistical and other properties of Riemann zeros based on an explicit equation for the $n$-th zero on the critical line
• An electrostatic depiction of the validity of the Riemann Hypothesis and a formula for the N-th zero at large N
• Order and Chaos in some deterministic infinite trigonometric products
• Randomness of Mobius coefficents and brownian motion: growth of the Mertens function and the Riemann Hypothesis
• The Coprime Quantum Chain
• Maximally Chaotic Dynamical Systems and Fundamental Interactions
• Quantum graphs whose spectra mimic the zeros of the Riemann zeta function
• Distribution of xp in some molecular rotational states
• Identifying primes from entanglement dynamics
• Generalized Riemann Hypothesis and Stochastic Time Series
• Quantum simulation of the Riemann-Hurwitz zeta function
• Generalized Riemann Hypothesis, Time Series and Normal Distributions
• Riemann zeta zeros and zero-point energy
• Supersymmetric quantum mechanics and the Riemann hypothesis
• Quantum simulation of the integer factorization problem: Bell states in a Penning trap
• Quantum Simulation of the Factorization Problem
• On the Emergence of the Quanta Prime Sequence
• Riemann zeros as quantized energies of scattering with impurities
• A lattice gas of prime numbers and the Riemann Hypothesis
• Application of abelian holonomy formalism to the elementary theory of numbers
• Phenomenological formula for Quantum Hall resistivity based on the Riemann zeta function
• Using quantum computers to identify prime numbers via entanglement dynamics
• A symmetry perspective of the Riemann zeros
• Functional equations for regularized zeta-functions and diffusion processes
• Physical interpretation of the Riemann hypothesis
• Quantum chaos in one dimension?
• Some results on natural numbers represented by quadratic polynomials in two variables
• Simple wave-optical superpositions as prime number sieves
• A Fowler-Nordheim Integrator can Track the Density of Prime Numbers
• Computing Riemann zeros with light scattering
• Hamiltonian with Energy Levels Corresponding to Riemann Zeros
• An intracellular calcium frequency code model extended to the Riemann zeta function
• Prime Number Identification Demonstrated with Quantum Processors Using a New Rescaling-Based Noise Mitigation Technique
• Low-dimensional Heisenberg magnets: Riemann zeta function regularization
• Achieving quantum advantage in a search for a violations of the Goldbach conjecture, with driven atoms in tailored potentials
• The Laplacian of The Integral Of The Logarithmic Derivative of the Riemann-Siegel-Hardy Z-function
• New generating and counting Functions of prime numbers applied to approximate Chebyschev 2nd class function and the least action principle applied to find non-trivial roots of the Zeta function and to Riemann Hypothesis
• Divergent Integrals of Certain Analytical Functions In the Sense of Zeta Regularization
• Quantum Counterpart of Equipartition Theorem: A Möbius Inversion Approach
• Life, The Mind, and Everything
• Analytic Continuation of Divergent Integrals
• Number-theoretic expressions obtained through analogy between prime factorization and optical interferometry
• Maximally chaotic dynamical systems of Anosov-Kolmogorov
• Equivalence between the zero distributions of the Riemann zeta function and a two-dimensional Ising model with randomly distributed competing interactions
• An optical Eratosthenes' sieve for large prime numbers