## recently archived material[Items are added to the top of this list as they are archived elsewhere.]

H. Matsui and Y. Matsumoto, "Revisiting regularization with Kaluza–Klein states and Casimir vacuum energy from extra dimensional spacetime" (preprint 04/2018)

[abstract:] "In the present paper, we investigate regularization of the one-loop quantum corrections with infinite Kaluza–Klein (KK) states and evaluate Casimir vacuum energy from extra dimensions. The extra dimensional models always involve the infinite massless or massive KK states, and therefore, the regularization of the infinite KK corrections is highly problematic. In order to avoid the ambiguity, we adopt the proper time integrals and the Riemann zeta function regularization in evaluating the summations of infinite KK states. In the calculation, we utilized the dimensional regularization method without exchanging the summations and the loop integrals. At the same time, we also evaluate the correction by the KK regularization method. Then, we clearly show that the regularized Casimir corrections from the KK states have the form of $1/R^2$ for the Higgs mass and $1/R^4$ for the cosmological constant, where $R$ is the compactification radius. We also evaluate the Casimir energy in supersymmetric extra-dimensional models. The contributions from bulk fermions and bulk bosons are not offset because the general boundary conditions break the supersymmetry. The non-zero supersymmetric Casimir corrections from extra dimensions undoubtedly contribute to the Higgs mass and the cosmological constant. We conclude such corrections are enhanced compared to the case without bulk supersymmetry."

R. Nally, "Exact half-BPS black hole entropies in CHL models from Rademacher series" (preprint 03/2018)

[abstract:] "The microscopic spectrum of half-BPS excitations in toroidally compactified heterotic string theory has been computed exactly through the use of results from analytic number theory. Recently, similar quantities have been understood macroscopically by evaluating the gravitational path integral on the M-theory lift of the AdS2 near-horizon geometry of the corresponding black hole. In this paper, we generalize these results to a subset of the CHL models, which include the standard compactification of IIA on $K3 \times T2$ as a special case. We begin by developing a Rademacher-like expansion for the Fourier coefficients of the partition functions for these theories, which are modular forms for congruence subgroups. We then interpret these results in a macroscopic setting by evaluating the path integral for the reduced-rank $N=4$ supergravities described by these CFTs."

M.L. Lapidus, "An overview of complex fractal dimensions: From fractal strings to fractal drums, and back" (preprint, 03/2018)

[abstract:] "Our main goal in this long survey article is to provide an overview of the theory of complex fractal dimensions and of the associated geometric or fractal zeta functions, first in the case of fractal strings (one-dimensional drums with fractal boundary), and then in the higher-dimensional case of relative fractal drums and, in particular, of arbitrary bounded subsets of Euclidean space of RN, for any integer $N=1$. Special attention is paid to discussing a variety of examples illustrating the general theory rather than to providing complete statements of the results and their proofs. Finally, in an epilogue, entitled "From quantized number theory to fractal cohomology", we briefly survey aspects of related work (motivated in part by the theory of complex fractal dimensions) of the author with H. Herichi (in the real case), and with T. Cobler (in the complex case), respectively, as well as in the latter part of a book in preparation by the author."

M.V. Berry, "Three quantum obsessions", Nonlinearity 21 (2008) T19–T26

[abstract:] "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?"

M.C. Nucci, "Spectral realization of the Riemann zeros by quantizing $H = w(x)(p + \ell_2_p/p)$: the Lie–Noether symmetry approach", Journal of Physics: Conference Series 482 (2014)

[abstract:] "If $t_n$ are the heights of the Riemann zeros $1/2 + it_n$, an old idea, attributed to Hilbert and Pólya, stated that the Riemann hypothesis would be proved if the $t_n$ could be shown to be eigenvalues of a self-adjoint operator. In 1986 Berry conjectured that $t_n$ could instead be the eigenvalues of a deterministic quantum system with a chaotic classical counterpart and in 1999 Berry and Keating 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 and therefore a more general Hamiltonian $H = w(x)(p + \ell_2p/p)$ was proposed and different expressions of the function $w(x)$ were considered although none of them yielding exactly $t_n$. We show that the quantization by means of Lie and Noether symmetries of the Lagrangian equation corresponding to the Hamiltonian $H$ yields straightforwardly the Schröodinger 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."

G.A.P. Ribeiro and A. Klümper, "Correlation functions of the integrable SU(n) spin chain" (preprint 04/2018)

[abstract:] "We study the correlation functions of $SU(n)$ $n>2$ invariant spin chains in the thermodynamic limit. We formulate a consistent framework for the computation of short-range correlation functions via functional equations which hold even at finite temperature. We give the explicit solution for two- and three-site correlations for the $SU(3)$ case at zero temperature. The correlators do not seem to be of factorizable form. From the two-sites result we see that the correlation functions are given in terms of Hurwitz' zeta function, which differs from the $SU(2)$ case where the correlations are expressed in terms of Riemann's zeta function of odd arguments."

Y. Abe, "Abelian Chern–Simons theory on the torus and physical views on the Hecke operators" (preprint 04/2018)

[abstract:] "In the previous paper [arXiv:1711.07122], we show that a holomorphic zero-mode wave function in abelian Chern–Simons theory on the torus can be considered as a quantum version of a modular form of weight 2. Motivated by this result, in this paper we consider an action of a Hecke operator on such a wave function from a gauge theoretic perspective. This leads us to obtain some physical views on the Hecke operators in number theory."

[abstract:] "In this paper, in Sections 1 and 2, we have described some equations and theorems concerning and linked to the Riemann zeta function. In the Section 3, we have showed the fundamental equation of the Riemann zeta function and the some equations concerning a new possible method for the calculation of the prime numbers. In conclusion, in the Section 4 we show the possible mathematical connections with various expressions of some sectors of string theory and number theory and finally we suppose as the prime numbers can be identified as possible solutions to the some equations of the string theory (zeta string)."

K. Pelka, J. Graf, T. Mehringer and J. von Zanthier, "Prime number decomposition using the Talbot effect" (preprint 03/2018)

[abstract:] "We report on prime number decomposition by use of the Talbot effect, a well-known phenomenon in classical near field optics whose description is closely related to Gauss sums. The latter are a mathematical tool from number theory used to analyze the properties of prime numbers as well as to decompose composite numbers into their prime factors. We employ the well-established connection between the Talbot effect and Gauss sums to implement prime number decompositions with a novel approach, making use of the longitudinal intensity profile of the Talbot carpet. The new algorithm is experimentally verified and the limits of the approach are discussed."

R. Bergamin, "$tt^*$ Geometry of modular curves" (preprint 03/2018)

[abstract:] "Motivated by Vafa's model, we study the $tt^{*}$ geometry of a degenerate class of FQHE models with an abelian group of symmetry acting transitively on the classical vacua. Despite it is not relevant for the phenomenology of the FQHE, this class of theories has interesting mathematical properties. We find that these models are parametrized by the family of modular curves $Y_{1}(N)= \mathbb{H}/\Gamma_{1}(N)$, labelled by an integer $N\geq 2$. Each point of the space of level $N$ is in correspondence with a one dimensional $\mathcal{N}=4$ Landau‐Ginzburg theory, which is defined on an elliptic curve with $N$ vacua and $N$ poles in the fundamental cell. The modular curve $Y(N)= \mathbb{H}/\Gamma(N)$ is a cover of degree $N$ of $Y_{1}(N)$ and plays the role of spectral cover for the space of models. The presence of an abelian symmetry allows to diagonalize the Berry's connection of the vacuum bundle and the $tt^{*}$ equations turn out to be the well known $\hat{A}_{N-1}$ Toda equations. The underlying structure of the modular curves and the connection between geometry and number theory emerge clearly when we study the modular properties and classify the critical limits of these models."

J. Ryu, M. Marciniak, M. Wieśniak, D. Kaszlikowski and M. Zukowski, "Entanglement conditions involving intensity correlations of optical fields: The case of multi-port interferometry" (preprint 03/2018)

[abstract:] "Normalized quantum Stokes operators introduced in [Phys. Rev. A 95, 042113 (2017)] enable one to better observe non-classical correlations of entangled states of optical fields with undefined photon numbers. For a given run of an experiment the new quantum Stokes operators are defined by the differences of the measured intensities (or photon numbers) at the exits of a polarizer divided by their sum. It is this ratio that is to be averaged, and not the numerator and the denominator separately, as it is in the conventional approach. The new approach allows to construct more robust entanglement indicators against photon-loss noise, which can detect entangled optical states in situations in which witnesses using standard Stokes operators fail. Here we show an extension of this approach beyond phenomena linked with polarization. We discuss EPR-like experiments involving correlations produced by optical beams in a multi-mode bright squeezed vacuum state. EPR-inspired entanglement conditions for all prime numbers of modes are presented. The conditions are much more resistant to noise due to photon loss than similar ones which employ standard Glauber-like intensity, correlations."

S. Torquato, G. Zhang and M. de Courcy-Ireland, "Uncovering multiscale order in the prime numbers via scattering" (preprint 02/2018)

[abstract:] "The prime numbers have been a source of fascination for millenia and continue to surprise us. Motivated by the hyperuniformity concept, which has attracted recent attention in physics and materials science, we show that the prime numbers in certain large intervals possess unanticipated order across length scales and represent the first example of a new class of many-particle systems with pure point diffraction patterns, which we call effectively limit-periodic. In particular, the primes in this regime are hyperuniform. This is shown analytically using the structure factor $S(k)$, proportional to the scattering intensity from a many-particle system. Remarkably, the structure factor for primes is characterized by dense Bragg peaks, like a quasicrystal, but positioned at certain rational wavenumbers, like a limit-periodic point pattern. We identify a transition between ordered and disordered prime regimes that depends on the intervals studied. Our analysis leads to an algorithm that enables one to predict primes with high accuracy. Effective limit-periodicity deserves future investigation in physics, independent of its link to the primes."

R.V. Ramos, "Quantum physics, algorithmic information theory and the Riemanns hypothesis" (preprint 12/2017)

[abstract:] "In the present work the Riemann hypothesis (RH) is discussed from four different perspectives. In the first case, coherent states and the Stengers approximation to Riemann-zeta function are used to show that RH avoids an indeterminacy of the type 0/0 in the inner product of two coherent states. In the second case, the Hilbert-Pólya conjecture with a quantum circuit is considered. In the third case, randomness, entanglement and the Moebius function are used to discuss the RH. At last, in the fourth case, the RH is discussed by inverting the first derivative of the Chebyshev function. The results obtained reinforce the belief that the RH is true."

G. Zhang, F. Martelli and S. Torquato, "Structure factor of the primes" (preprint 01/2018)

[abstract:] "Although the prime numbers are deterministic, they can be viewed, by some measures, as pseudo-random numbers. In this article, we numerically study the pair statistics of the primes using statistical-mechanical methods, especially the structure factor $S(k)$ in an interval $M=p=M+L$ with $M$ large, and $L/M$ smaller than unity. We show that the structure factor of the prime configurations in such intervals exhibits well-defined Bragg-like peaks along with a small "diffuse" contribution. This indicates that the primes are appreciably more correlated and ordered than previously thought. Our numerical results definitively suggest an explicit formula for the locations and heights of the peaks. This formula predicts infinitely many peaks in any non-zero interval, similar to the behavior of quasicrystals. However, primes differ from quasicrystals in that the ratio between the location of any two predicted peaks is rational. We also show numerically that the diffuse part decays slowly as $M$ or $L$ increases. This suggests that the diffuse part vanishes in an appropriate infinite-system-size limit."

J. Peng, S. Sun, V.K. Narayana, V.J. Sorger and T. El-Ghazawi, Integrated nanophotonics architecture for residue number system arithmetic" (preprint 11/2017)

[abstract:] "Residue number system (RNS) enables dimensionality reduction of an arithmetic problem by representing a large number as a set of smaller integers, where the number is decomposed by prime number factorization using the moduli as basic functions. These reduced problem sets can then be processed independently and in parallel, thus improving computational efficiency and speed. Here we show an optical RNS hardware representation based on integrated nanophotonics. The digit-wise shifting in RNS arithmetic is expressed as spatial routing of an optical signal in $2\times 2$ hybrid photonic-plasmonic switches. Here the residue is represented by spatially shifting the input waveguides relative to the routers outputs, where the moduli are represented by the number of waveguides. By cascading the photonic $2\times 2$ switches, we design a photonic RNS adder and a multiplier forming an all-to-all sparse directional network. The advantage of this photonic arithmetic processor is the short (10's ps) computational execution time given by the optical propagation delay through the integrated nanophotonic router. Furthermore, we show how photonic processing in-the-network leverages the natural parallelism of optics such as wavelength-division-multiplexing or optical angular momentum in this RNS processor. A key application for photonic RNS is the functional analysis convolution with widespread usage in numerical linear algebra, computer vision, language- image- and signal processing, and neural networks."

J. Sakhr and J.M. Nieminen, "Local box-counting dimensions of discrete quantum eigenvalue spectra: Analytical connection to quantum spectral statistics" (preprint 11/2017)

[abstract:] "Two decades ago, Wang and Ong [Phys. Rev. A 55, 1522 (1997)] hypothesized that the local box-counting dimension of a discrete quantum spectrum should depend exclusively on the nearest-neighbor spacing distribution (NNSD) of the spectrum. In this paper, we validate their hypothesis by deriving an explicit formula for the local box-counting dimension of a countably-infinite discrete quantum spectrum. This formula expresses the local box-counting dimension of a spectrum in terms of single and double integrals of the NNSD of the spectrum. As applications, we derive an analytical formula for Poisson spectra and closed-form approximations to the local box-counting dimension for spectra having Gaussian orthogonal ensemble (GOE), Gaussian unitary ensemble (GUE), and Gaussian symplectic ensemble (GSE) spacing statistics. In the Poisson and GOE cases, we compare our theoretical formulas with the published numerical data of Wang and Ong and observe excellent agreement between their data and our theory. We also study numerically the local box-counting dimensions of the Riemann zeta function zeros and the alternate levels of GOE spectra, which are often used as numerical models of spectra possessing GUE and GSE spacing statistics, respectively. In each case, the corresponding theoretical formula is found to accurately describe the numerically-computed local box-counting dimension."

R.S. Mackay, "Towards a spectral proof of Riemann's hypothesis" (preprint 08/2017)

[abstract:] "The paper presents evidence that Riemann's $\xi$ function evaluated at $2\sqrt(E)$ could be the characteristic function $P(E)$ for the magnetic Laplacian minus $85/16$ on a surface of curvature $-1$ with magnetic field $9/4$, a cusp of width $1$, a Dirichlet condition at a point, and other conditions not yet determined."

[abstract:] "Argument principle calculations of zeroes using extended Riemann Siegel function analogues are investigated for the Davenport-Heilbronn function and another 5-periodic Dirichlet series. Both these 5-periodic Dirichlet series functions have previously been reported as possessing functional equations and not being expressible as Euler products. Informative counts of the non-trivial zeroes (including off the critical line) are obtained for various $s = \sigma + it$ lines in the positive quadrant of the complex plane."

"Perspectives on the Riemann Hypothesis", School of Mathematics, University of Bristol, 4–7 June 2018

"A meeting on the Riemann Hypothesis and on the theory of the zeta-function and other L-functions."

Diophantine Approximation and Dynamical Systems", La Trobe University, Melbourne, 6–8 January 2018

A. Donis-Vela and J.C. Garcia-Escartin, "A quantum primality test with order finding" (preprint 11/2017)

[abstract:] "Determining whether a given integer is prime or composite is a basic task in number theory. We present a primality test based on quantum order finding and the converse of Fermat's theorem. For an integer $N$, the test tries to find an element of the multiplicative group of integers modulo $N$ with order $N-1$. If one is found, the number is known to be prime. During the test, we can also show most of the times $N$ is composite with certainty (and a witness) or, after $\log\log N$ unsuccessful attempts to find an element of order $N-1$, declare it composite with high probability. The algorithm requires $O((\log n)^2n^3)$ operations for a number $N$ with $n$ bits, which can be reduced to $O(\log\log n(\log n)^3n^2)$ operations in the asymptotic limit if we use fast multiplication."

Y. Chen, A. Prakash and T.-C. Wei, "Universal quantum computing using $(\mathbb_d)^3$ symmetry-protected topologically ordered states" (preprint 11/2017)

[abstract:] "Measurement-based quantum computation describes a scheme where entanglement of resource states is utilized to simulate arbitrary quantum gates via local measurements. Recent works suggest that symmetry-protected topologically non-trivial, short-ranged entanged states are promising candidates for such a resource. Miller and Miyake [NPJ Quantum Information 2, 16036 (2016)] recently constructed a particular $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2\times$ symmetry-protected topological state on the Union-Jack lattice and established its quantum computational universality. However, they suggested that the same construction on the triangular lattice might not lead to a universal resource. Instead of qubits, we generalize the construction to qudits and show that the resulting $(d-1)$ qudit nontrivial $\mathbb{Z}_2\times \mathbb{Z}_2\times \mathbb{Z}_2\times$ symmetry-protected topological states are universal on the triangular lattice, for $d$ being a prime number greater than $2$. The same construction also holds for other $3$-colorable lattices, including the Union-Jack lattice."

A. Roussou, J. Smyrnakis, M. Magiropoulos, Nikolaos K. Efremidis, G.M. Kavoulakis, P. Sandin, M. Ögren and M. Gulliksson, "Excitation spectrum of a mixture of two Bose gases confined in a ring potential with interaction asymmetry" (preprint 11/2017)

[abstract:] "We study the rotational properties of a two-component Bose–Einstein condensed gas of distinguishable atoms which are confined in a ring potential using both the mean-field approximation, as well as the method of diagonalization of the many-body Hamiltonian. We demonstrate that the angular momentum may be given to the system either via single-particle, or "collective" excitation. Furthermore, despite the complexity of this problem, under rather typical conditions the dispersion relation takes a remarkably simple and regular form. Finally, we argue that under certain conditions the dispersion relation is determined via collective excitation. The corresponding many-body state, which, in addition to the interaction energy minimizes also the kinetic energy, is dictated by elementary number theory."

J. Benatar, D. Marinucci and I. Wigman, "Planck-scale distribution of nodal length of arithmetic random waves" (preprint 11/2017)

[abstract:] "We study the nodal length of random toral Laplace eigenfunctions ("arithmetic random waves") restricted to decreasing domains ("shrinking balls"), all the way down to Planck scale. We find that, up to a natural scaling, for "generic" energies the variance of the restricted nodal length obeys the same asymptotic law as the total nodal length, and these are asymptotically fully correlated. This, among other things, allows for a statistical reconstruction of the full toral length based on partial information. One of the key novel ingredients of our work, borrowing from number theory, is the use of bounds for the so-called spectral Quasi-Correlations, i.e. unusually small sums of lattice points lying on the same circle."

C.M. Bender and D.C. Brody, "Asymptotic analysis on a pseudo-Hermitian Riemann-zeta Hamiltonian" (preprint 11/2017)

"The differential equation eigenvalue problem associated with a recently-introduced Hamiltonian, whose eigenvalues correspond to the zeros of the Riemann zeta function, is analyzed using the Fourier and the WKB analysis. The Fourier analysis leads to an open problem concerning the formulation of the eigenvalue problem in the momentum space. The WKB analysis gives an exact asymptotic behavior of the eigenfunction."

A. Lipniacka and B.M. Dit Latour, "A quantum mechanical well and a derivation of a $\pi^2$ formula" (preprint 11/2017)

[abstract:] "Quantum particle bound in an infinite, one-dimensional square potential well is one of the problems in Quantum Mechanics (QM) that most of the textbooks start from. There, calculating an allowed energy spectrum for an arbitrary wave function often involves Riemann zeta function resulting in a $\pi$ series. In this work, two "$\pi$ formulas" are derived when calculating a spectrum of possible outcomes of the momentum measurement for a particle confined in such a well, the series, $\frac{\pi^2}{8} = \sum_{k=1}^{k=\infty} \frac{1}{(2k-1)^2}$, and the integral $\int_{-\infty}^{\infty} \frac{sin^2 x}{x^2} dx =\pi$. The spectrum of the momentum operator appears to peak on classically allowed momentum values only for the states with even quantum number. The present article is inspired by another quantum mechanical derivation of $\pi$ formula in \cite{wallys}."