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

Number Theory and Dynamics, March 25–29, 2019, University of Cambridge, UK

K. Nakayama, F. Takahashi and T.T. Yanagida, "Revisiting the number-theory dark matter scenario and the weak gravity conjecture" (preprint 11/2018)

[abstract:] "We revisit the number-theory dark matter scenario where one of the light chiral fermions required by the anomaly cancellation conditions of $U(1)_{B-L}$ explains dark matter. Focusing on some of the integer B-L charge assignments, we explore a new region of the parameter space where there appear two light fermions and the heavier one becomes a dark matter of mass O(10)keV or O(10)MeV. The dark matter radiatively decays into neutrino and photon, which can explain the tantalizing hint of the 3.55keV X-ray line excess. Interestingly, the other light fermion can erase the AdS vacuum around the neutrino mass scale in a compactification of the standard model to 3D. This will make the standard model consistent with the AdS-WGC statement that stable non-supersymmetric AdS vacua should be absent."

M. Bishop, E. Aiken and D. Singleton, "Modified commutation relationships from the Berry–Keating program" (preprint 10/2018)

[abstract:] "Current approaches to quantum gravity suggest there should be a modification of the standard quantum mechanical commutator, $[\hat{\mathbf{x}},\hat{\mathbf{p}}] = i\hbar$. Typical modifications are phenomenological and designed to result in a minimal length scale. As a motivating principle for the modification of the position and momentum commutator, we assume the validity of a version of the Bender–Brody–Müller variant of the Berry–Keating approach to the Riemann hypothesis. We arrive at a family of modified position and momentum operators, and their associated modified commutator, which lead to a minimal length scale. Additionally, this larger family generalizes the Bender–Brody–Müller approach to the Riemann hypothesis."

L. Kocia and P. Love, "Stationary phase method in discrete Wigner functions and classical simulation of quantum circuits" (preprint 10/2018)

[abstract:] "We apply the periodized stationary phase method to discrete Wigner functions of systems with odd prime dimension using results from $p$-adic number theory. We derive the Wigner–Weyl–Moyal (WWM) formalism with higher order $\hbar$ corrections representing contextual corrections to non-contextual Clifford operations. We apply this formalism to a subset of unitaries that include diagonal gates such as the $\frac{\pi}{8}$ gates. We characterize the stationary phase critical points as a quantum resource injecting contextuality and show that this resource allows for the replacement of the $p^{2t}$ points that represent $t$ magic state Wigner functions on $p$-dimensional qudits by $\leq p^t$ points. We find that the $\frac{\pi}{8}$ gate introduces the smallest higher order $\hbar$ correction possible, requiring the lowest number of additional critical points compared to the Clifford gates. We then establish a relationship between the stabilizer rank of states and the number of critical points necessary to treat them in the WWM formalism. This allows us to exploit the stabilizer rank decomposition of two qutrit $\frac{\pi}{8}$ gates to develop a classical strong simulation of a single qutrit marginal on $t$ qutrit $\frac{\pi}{8}$ gates that are followed by Clifford evolution, and show that this only requires calculating $3^{\frac{t}{2}+1}$ critical points corresponding to Gauss sums. This outperforms the best alternative qutrit algorithm (based on Wigner negativity and scaling as $\approx 3^{0.8t}$ for $10^{-2}$ precision) for any number of $\frac{\pi}{8}$ gates to full precision."

D.C. Brody and C.M. Bender, "Operator-valued zeta functions and Fourier analysis" (preprint 10/2018)

[abstract:] "The Riemann zeta function $\zeta(s)$ is defined as the infinite sum $\sum_{n=1}^{\infty}n^{-s}$, which converges when $\Re s>1$. The Riemann hypothesis asserts that the nontrivial zeros of $\zeta(s)$ lie on the line $\Re s = \frac{1}{2}$. Thus, to find these zeros it is necessary to perform an analytic continuation to a region of complex $s$ for which the defining sum does not converge. This analytic continuation is ordinarily performed by using a functional equation. In this paper it is argued that one can investigate some properties of the Riemann zeta function in the region $\Re s<1$ by allowing operator-valued zeta functions to act on test functions. As an illustration, it is shown that the locations of the trivial zeros can be determined purely from a Fourier series, without relying on an explicit analytic continuation of the functional equation satisfied by $\zeta(s)$."

S.C.L. Srivastava, A. Lakshminarayan, S. Tomsovic and A. Bäcker, "Ordered level spacing probability densities" (preprint 10/2018)

[abstract:] "Spectral statistics of quantum systems have been studied in detail using the nearest neighbour level spacings, which for generic chaotic systems follows random matrix theory predictions. In this work, the probability density of the closest neighbour and farther neighbour spacings from a given level are introduced. Analytical predictions are derived using a $3\times 3$ matrix model. The closest neighbour density is generalized to the $k$-th closest neighbour spacing density, which allows for investigating long-range correlations. For larger $k$ the probability density of $k$-th closest neighbour spacings is well described by a Gaussian. Using these $k$-th closest neighbour spacings we propose the ratio of the closest neighbour to the second closest neighbour as an alternative to the ratio of successive spacings. For a Poissonian spectrum the density of the ratio is flat, whereas for the three Gaussian ensembles repulsion at small values is found. The ordered spacing statistics and their ratio are numerically studied for the integrable circle billiard, the chaotic cardioid billiard, the standard map and the zeroes of the Riemann zeta function. Very good agreement with the predictions is found."

A. LeClair and G. Mussardo, "Generalized Riemann Hypothesis, time series and normal distributions" (preprint 09/2018)

[abstract:] "$L$ functions based on Dirichlet characters are natural generalizations of the Riemann $\zeta(s)$ function: they both have series representations and satisfy an Euler product representation, i.e. an infinite product taken over prime numbers. In this paper we address the Generalized Riemann Hypothesis relative to the non-trivial complex zeros of the Dirichlet $L$ functions by studying the possibility to enlarge the original domain of convergence of their Euler product. The feasibility of this analytic continuation is ruled by the asymptotic behavior in $N$ of the series $B_N = \sum_{n=1}^N\cos(t\logp_n-\arg\chi(p_n))$ involving Dirichlet characters $\chi$ modulo $q$ on primes $p^n$. Although deterministic, these series have pronounced stochastic features which make them analogous to random time series. We show that the $B_N$'s satisfy various normal law probability distributions. The study of their large asymptotic behavior poses an interesting problem of statistical physics equivalent to the Single Brownian Trajectory Problem, here addressed by defining an appropriate ensemble $\mathcal{E}$ involving intervals of primes. For non-principal characters, we show that the series $B_N$ present a universal diffusive random walk behavior $B_N = O(\sqrt{N})$ in view of the Dirichlet theorem on the equidistribution of reduced residue classes modulo $q$ and the Lemke Oliver–Soundararajan conjecture on the distribution of pairs of residues on consecutive primes. This purely diffusive behavior of $B_N$ implies that the domain of convergence of the infinite product representation of the Dirichlet $L$-functions for non-principal characters can be extended from $\Re(s) > 1$ down to $\Re(s) = 1/2$, without encountering any zeros before reaching this critical line."

A.H. Chamseddine, A. Connes and W.D. van Suijlekom, "Entropy and the spectral action" (preprint 09/2018)

[abstract:] "We compute the information theoretic von Neumann entropy of the state associated to the fermionic second quantization of a spectral triple. We show that this entropy is given by the spectral action of the spectral triple for a specific universal function. The main result of our paper is the surprising relation between this function and the Riemann zeta function. It manifests itself in particular by the values of the coefficients $c(d)$ by which it multiplies the $d$ dimensional terms in the heat expansion of the spectral triple. We find that $c(d)$ is the product of the Riemann xi function evaluated at $-d$ by an elementary expression. In particular $c(4)$ is a rational multiple of $\zeta(5)$ and $c(2)$ a rational multiple of $\zeta(3)$. The functional equation gives a duality between the coefficients in positive dimension, which govern the high energy expansion, and the coefficients in negative dimension, exchanging even dimension with odd dimension."

P. Braun and D. Waltner, "New approach to periodic orbit theory of spectral correlations" (preprint 09/2018)

[abstract:] "The existing periodic orbit theory of spectral correlations for classically chaotic systems relies on the Riemann–Siegel-like representation of the spectral determinants which is still largely hypothetical. We suggest a simpler derivation using analytic continuation of the periodic-orbit expansion of the pertinent generating function from the convergence border to physically important real values of its arguments. As examples we consider chaotic systems without time reversal as well as the Riemann zeta function and Dirichlet $L$-functions zeros."

Eminent mathematical physicist Sir Michael Atiyah presented a purported "simple proof" of the RH at a conference in Heidelberg on 24th September 2018. It seems that he arrived at this as an unexpected "bonus" when attempting to derive the fine structure constant. Details are still coming in, but it seems that the key ingredient is the Todd function. Here is Atiyah's paper, and here is another one it heavily references. Here is a video of Atiyah's talk. Reactions thus far have been predominantly sceptical – see, e.g., this. Check back here for updates.

R.J. Lipton, "Reading into Atiyah's Proof"

I. Perito, A. Roncaglia and A. Bendersky, "Selective and Efficient Quantum Process Tomography in Arbitrary Finite Dimension" (preprint 08/2018)

[abstract:] "The characterization of quantum processes is a key tool in quantum information processing tasks for several reasons: on one hand, it allows to acknowledge errors in the implementations of quantum algorithms; on the other, it allows to charcaterize unknown processes occurring in Nature. In previous works [Bendersky, Pastawski and Paz. Phys. Rev. Lett. 100, 190403 (2008) and Phys. Rev. A 80, 032116 (2009)] it was introduced a method to selectively and efficiently measure any given coefficient from the matrix description of a quantum channel. However, this method heavily relies on the construction of maximal sets of mutually unbiased bases (MUBs), which are known to exist only when the dimension of the Hilbert space is the power of a prime number. In this article, we lift the requirement on the dimension by presenting two variations of the method that work on arbitrary finite dimensions: one uses tensor products of maximally sets of MUBs, and the other uses a dimensional cutoff of a higher prime power dimension."

A. Chattopadhyay, P. Dutta, S. Dutta and D. Ghoshal, "Matrix model for Riemann zeta via its local Factors" (preprint 07/2018)

[abstract:] "We propose the construction of an ensemble of unitary random matrices (UMM) for the Riemann zeta function. Our approach to this problem is 'piecemeal', in the sense that we consider each factor in the Euler product representation of the zeta function to first construct a UMM for each prime $p$. We are able to use its phase space description to write the partition function as the trace of an operator that acts on a subspace of square-integrable functions on the $p$-adic line. This suggests a Berry–Keating type Hamiltonian. We combine the data from all primes to propose a Hamiltonian and a matrix model for the Riemann zeta function."

J. Kaniewski, I. Supić, J. Tura, F. Baccari, A. Salavrakos and R. Augusiak, "Maximal nonlocality from maximal entanglement and mutually unbiased bases, and self-testing of two-qutrit quantum systems (preprint 07/2018)

[abstract:] "Bell inequalities are an important tool in device-independent quantum information processing because their violation can serve as a certificate of relevant quantum properties. Probably the best known example of a Bell inequality is due to Clauser, Horne, Shimony and Holt (CHSH), defined in the simplest scenario involving two dichotomic measurements, whose all key properties are well understood. While there have been many attempts to generalise it to higher-dimensional quantum systems, quite surprisingly, most of them turn out to be difficult to analyse. In particular, the maximal quantum violation—a key quantity for most device-independent applications—remains unknown except for the simplest cases. Here we propose a new generalisation of the CHSH Bell inequality which preserves several of its attractive features: the maximal quantum value can be computed analytically and can be achieved by the maximally entangled states and mutually unbiased bases. These inequalities involve $d$ measurements settings, each having $d$ outcomes for an arbitrary prime number $d\geq 3$. We then show that in the three-outcome case our Bell inequality is a self-test: it can be used to self-test the maximally entangled state of two-qutrits and three mutually unbiased bases at each site. Yet, we demonstrate that in the case of more outcomes, their maximal violation does not allow for self-testing in the standard sense, which suggests a new weak form of self-testing. The ability to certify high-dimensional MUBs makes them attractive from the device-independent cryptography point of view."

N. Benjamin, S. Kachru, K. Ono and L. Rolen, "Black holes and class groups" (preprint 07/2018)

[abstract:] "The theory of quadratic forms and class numbers has connections to many classical problems in number theory. Recently, class numbers have appeared in the study of black holes in string theory. We describe this connection and raise questions in the hope of inspiring new collaborations between number theorists and physicists."

A. Bernal, "On the existence of absolutely maximally entangled states of minimal support II" (preprint 07/2018)

[abstract:] "Absolutely maximally entangled, AME, states are pure multipartite states that give rise to the maximally mixed states when half or more of the parties are traced out. AME states find applications in fields like teleportation or quantum secret sharing, and the problem of finding conditions on their existence has been considered in a number of papers. We consider here AME states of minimal support, that are simpler to analyse. An equivalence with coding theory gives a sufficient condition for their existence, that the number of sites be equal to the local dimension plus one, when the local dimension $d$ is a power of a prime number. In this paper, an equivalence with latin hypercubes is used to prove that the above sufficient condition fails in the first case in which the local dimension is not a prime power, $d=6$. Results for other values of $d$ are also given."

C.H.A. Cheng, R. Granero-Belinchon, S. Shkoller and J. Wilkening, "Rigorous asymptotic models of water waves" (preprint 07/2018)

"We develop a rigorous asymptotic derivation for two mathematical models of water waves that capture the full nonlinearity of the Euler equations up to quadratic and cubic interactions, respectively. Specifically, letting epsilon denote an asymptotic parameter denoting the steepness of the water wave, we use a Stokes expansion in epsilon to derive a set of linear recursion relations for the tangential component of velocity, the stream function, and the water wave parameterization. The solution of the water waves system is obtained as an infinite sum of solutions to linear problems at each $\epsilon^k$ level, and truncation of this series leads to our two asymptotic models, that we call the quadratic and cubic $h$-models. Using the growth rate of the Catalan numbers (from number theory), we prove well-posedness of the $h$-models in spaces of analytic functions, and prove error bounds for solutions of the $h$-models compared against solutions of the water waves system. We also show that the Craig–Sulem models of water waves can be obtained from our asymptotic procedure and that their WW2 model is well-posed in our functional framework. We then develop a novel numerical algorithm to solve the quadratic and cubic $h$-models as well as the full water waves system. For three very different examples, we show that the agreement between the model equations and the water waves solution is excellent, even when the wave steepness is quite large. We also present a numerical example of corner formation for water waves."

A. Varchenko, "Hyperelliptic integrals modulo $p$ and Cartier–Manin matrices" (preprint 06/2018)

[abstract:] "The hypergeometric solutions of the KZ equations were constructed almost 30 years ago. The polynomial solutions of the KZ equations over the finite field $F_p$ with a prime number $p$ of elements were constructed recently. In this paper we consider the example of the KZ equations whose hypergeometric solutions are given by hyperelliptic integrals of genus $g$. It is known that in this case the total $2g$-dimensional space of holomorphic solutions is given by the hyperelliptic integrals. We show that the recent construction of the polynomial solutions over the field $F_p$ in this case gives only a $g$-dimensional space of solutions, that is, a 'half' of what the complex analytic construction gives. We also show that all the constructed polynomial solutions over the field $F_p$ can be obtained by reduction modulo $p$ of a single distinguished hypergeometric solution. The corresponding formulas involve the entries of the Cartier–Manin matrix of the hyperelliptic curve. That situation is analogous to the example of the elliptic integral considered in the classical Y.I. Manin's paper in 1961."

S. Torquato, "Basic understanding of condensed phases of matter via packing models" (preprint, 05/2018)

[abstract:] "Packing problems have been a source of fascination for millenia and their study has produced a rich literature that spans numerous disciplines. Investigations of hard-particle packing models have provided basic insights into the structure and bulk properties of condensed phases of matter, including low-temperature states (e.g., molecular and colloidal liquids, crystals and glasses), multiphase heterogeneous media, granular media, and biological systems. The densest packings are of great interest in pure mathematics, including discrete geometry and number theory. This perspective reviews pertinent theoretical and computational literature concerning the equilibrium, metastable and nonequilibrium packings of hard-particle packings in various Euclidean space dimensions. In the case of jammed packings, emphasis will be placed on the 'geometric-structure' approach, which provides a powerful and unified means to quantitatively characterize individual packings via jamming categories and 'order' maps. It incorporates extremal jammed states, including the densest packings, maximally random jammed states, and lowest-density jammed structures. Packings of identical spheres, spheres with a size distribution, and nonspherical particles are also surveyed. We close this review by identifying challenges and open questions for future research."

H. Li, F. Li and S. Kanemits, Number Theory and Its Applications II (World Scientific, 2018)

W. Dittrich, Reassessing Riemann's paper on the number of primes less than a given magnitude (Springer, 2018)

H. Montgomery, A. Nikeghbali and M.T. Rassias, eds., Exploring the Riemann Zeta function (Springer 2017)

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}."