##
recently archived material
[Items are added to the top of this list as they are archived elsewhere.]
L.F. Alday and J.-B. Bae, "Rademacher expansions and the spectrum of 2d CFT" (preprint 12/2019)
[abstract:] "A classical result from analytic number theory by Rademacher gives an exact formula for the Fourier coefficients of modular forms of non-positive weight. We apply similar techniques to study the spectrum of two-dimensional unitary conformal field theories, with no extended chiral algebra and $c > 1$. By exploiting the full modular constraints of the partition function we propose an expression for the spectral density in terms of the light spectrum of the theory. The expression is given in terms of a Rademacher expansion, which converges for spin $j \neq 0$. For a finite number of light operators the expression agrees with a variant of the Poincare construction developed by Maloney, Witten and Keller. With this framework we study the presence of negative density of states in the partition function dual to pure gravity, and propose a scenario to cure this negativity."
[added 10th January 2020]
L. Dal Negro, Y. Chen and F. Sgrignuoli, "Aperiodic photonics of elliptic curves" (preprint 12/2019)
[abstract:] "In this paper we propose a novel approach to aperiodic order in optical science and technology that leverages the intrinsic structural complexity of certain non-polynomial (hard) problems in number theory and cryptography for the engineering of optical media with novel transport and wave localization properties. In particular, we address structure-property relationships in a large number (900) of light scattering systems that physically manifest the distinctive aperiodic order of elliptic curves and the associated discrete logarithm problem over finite fields. Besides defining an extremely rich subject with profound connections to diverse mathematical areas, elliptic curves offer unprecedented opportunities to engineer light scattering phenomena in aperiodic environments beyond the limitations of traditional random media. Our theoretical analysis combines the interdisciplinary methods of point patterns spatial statistics with the rigorous Green's matrix solution of the multiple wave scattering problem for electric and magnetic dipoles and provides access to the spectral and light scattering properties of novel deterministic aperiodic structures with enhanced light-matter coupling for nanophotonics and metamaterials applications to imaging and spectroscopy."
[added 10th January 2020]
M. Nardelli and A. Nardelli, "On Ramanujan's equations applied to various sectors of particle physics and cosmology: New possible mathematical connections, VII" (preprint 2020)
M. Nardelli and A. Nardelli, "Mathematical connections between various Ramanujan equations, values of mass and electric charges of fundamental particles and physical data of Kerr Supermassive Black Hole M87" (preprint 2020)
M. Nardelli and A. Nardelli, "A new possible theory of mathematical connections between some Ramanujan equations and approximations to $\pi$, the equations of inflationary cosmology concerning the scalar field $\phi$, the inflaton mass, the Higgs boson mass and the Pion meson $\pi^{\pm}$ mass" (preprint 2020)
[added 10th January 2020]
M. Nardelli and A. Nardelli, "On some new possible mathematical connections between some equations of the Ramanujan's manuscripts, the Rogers–Ramanujan continued fractions and some sectors of particle physics, string theory and D-branes" (preprint 10/19)
[abstract:] "In this research thesis, we have described some new mathematical connections between some equations of the Ramanujan's manuscripts, the Rogers–Ramanujan continued fractions and some sectors of particle physics (physical parameters of mesons and dilatons, in particular the values of the masses), string theory and D-branes."
M. Nardelli and A. Nardelli, "On the possible mathematical connections between some equations of various sectors concerning the D-Branes and some Ramanujan’s modular equations and approximations to $\pi$" (preprint 10/19)
M. Nardelli and A. Nardelli, "On the possible mathematical connections between some equations of various topics concerning the Dilaton value, the D-Brane, the Bouncing Cosmology and some sectors of number theory (Riemann's functions of S. Ramanujan and Rogers–Ramanujan continued fractions)" (preprint 10/19)
[added 30th October 2019]
B. Li, G. Maltese, J.I. Costa-Filho, A.A. Pushkina and A.I. Lvovsky, "An optical Eratosthenes' sieve for large prime numbers" (preprint 10/2019)
[abstract:] "We report the first experimental demonstration of prime number sieve via linear optics. The prime numbers distribution is encoded in the intensity zeros of the far field produced by a spatial light modulator hologram, which comprises a set of diffraction gratings whose periods correspond to all prime numbers below $149$. To overcome the limited far field illumination window and the discretization error introduced by the SLM finite spatial resolution, we rely on additional diffraction gratings and sequential recordings of the far field. This strategy allows us to optically sieve all prime numbers below $149^2 = 22201$."
[added 30th October 2019]
A. Connes, "Noncommutative Geometry, the spectral standpoint" (preprint 10/2019)
[abstract:] "We report on the following highlights from among the many discoveries made in Noncommutative Geometry since year 2000: 1) The interplay of the geometry with the modular theory for noncommutative tori, 2) Advances on the Baum–Connes conjecture, on coarse geometry and on higher index theory, 3) The geometrization of the pseudo-differential calculi using smooth groupoids, 4) The development of Hopf cyclic cohomology, 5) The increasing role of topological cyclic homology in number theory, and of the lambda operations in archimedean cohomology, 6) The understanding of the renormalization group as a motivic Galois group, 7) The development of quantum field theory on noncommutative spaces, 8) The discovery of a simple equation whose irreducible representations correspond to $4$-dimensional spin geometries with quantized volume and give an explanation of the Lagrangian of the standard model coupled to gravity, 9) The discovery that very natural toposes such as the scaling site provide the missing algebro-geometric structure on the noncommutative adele class space underlying the spectral realization of zeros of $L$-functions."
[added 30th October 2019]
T. Asselmeyer-Maluga, "Braids, $3$-manifolds, elementary particles: Number theory and symmetry in particle physics" (preprint 10/2019)
[abstract:] "In this paper, we will describe a topological model for elementary particles based on $3$-manifolds. Here, we will use Thurston's geometrization theorem to get a simple picture: fermions as hyperbolic knot complements (a complement $C(K) = S^3∖(K\times D^2)$ of a knot $K$ carrying a hyperbolic geometry) and bosons as torus bundles. In particular, hyperbolic $3$-manifolds have a close connection to number theory (Bloch group, algebraic $K$-theory, quaternionic trace fields), whichwill be used in the description of fermions. Here, we choose the description of $3$-manifolds by branched covers. Every $3$-manifold can be described by a $3$-fold branched cover of $S^3$ branched along a knot. In case of knot complements, one will obtain a $3$-fold branched cover of the $3$-disk $D^3$ branched along a $3$-braid or $3$-braids describing fermions. The whole approach will uncover new symmetries as induced by quantum and discrete groups. Using the Drinfeld–Turaev quantization, we will also construct a quantization so that quantum states correspond to knots. Particle properties like the electric charge must be expressed by topology, and we will obtain the right spectrum of possible values. Finally, we will get a connection to recent models of Furey, Stoica and Gresnigt using octonionic and quaternionic algebras with relations to $3$-braids (Bilson–Thompson model)."
[added 30th October 2019]
Y.-L. Wang, "
Special unextendible entangled bases with continuous integer cardinality" (preprint 09/2019)
[abstract:] "Special unextendible entangled basis of ``type $k$’’ (SUEBk), a set of incomplete orthonormal special entangled states of ``type $k$’’ whose complementary space has no special entangled state of ``type $k$’’. This concept can be seem as a generalization of the unextendible product basis (UPB) introduced by Bennett et al. in [ Phys. Rev. Lett. \textbf{82}, 5385(1999) ] and the unextendible maximally entangled basis (UMEB) introduced by Bravyi and Smolin in [Phys. Rev. A \textbf{84}, 042306(2011)]. We present an efficient method to construct sets of SUEBk. The main strategy here is to decompose the whole space into two subspaces such that the rank of one subspace can be easily upper bounded by $k$ while the other one can be generated by two kinds of the special entangled states of type $k$. This method is very effective for those $k = p^m \geq 3$ where $p$ is a prime number. For these cases, we can otain sets of SUEBk with continuous integer cardinality when the local dimensions are large. Moreover, one can find that our method here can be easily extended when there are more than two kinds of the special entangled states of type $k$ at hand."
[added 30th October 2019]
H. García-Compeán, E.Y. López and W.A. Zúñiga-Galindo, "p-Adic open string amplitudes with Chan–Paton factors coupled to a constant B-field" (preprint 09/2019)
[abstract:] "We establish rigorously the regularization of the $p$-adic open string amplitudes, with Chan–Paton rules and a constant B-field, introduced by Goshal and Kawano. In this study we use techniques of multivariate local zeta functions depending on multiplicative characters and a phase factor which involves an antisymmetric bilinear form. These local zeta functions are new mathematical objects. We attach to each amplitude a multivariate local zeta function depending on the kinematics parameters, the B-field and the Chan–Paton factors. We show that these integrals admit meromorphic continuations in the kinematic parameters, this result allows us to regularize the Goshal–Kawano amplitudes, the regularized amplitudes do not have ultraviolet divergences. Due to the need of a certain symmetry, the theory works only for prime numbers which are congruent to $3$ modulo $4$. We also discuss the limit $p$ tends to $1$ in the noncommutative effective field theory and in the Ghoshal–Kawano amplitudes. We show that in the case of four points, the limit $p$ tends to $1$ of the regularized Ghoshal–Kawano amplitudes coincides with the Feynman amplitudes attached to the limit $p$ tends to $1$ of the noncommutative Gerasimov –Shatashvili Lagrangian."
[added 30th October 2019]
A. Chávez, H. Prado and E. G. Reyes, "The Borel transform and linear nonlocal equations: Applications to zeta-nonlocal field models" (preprint 07/2019)
[abstract:] "We define rigorously operators of the form $f(\partial_t)$, in which $f$ is an analytic function on a simply connected domain. Our formalism is based on
the Borel transform on entire functions of exponential type. We study existence and regularity of real-valued solutions for the nonlocal in time equation
\begin{equation*}
f(\partial_t)\phi= J(t) \; \; , \quad t\in \mathbb{R}\; ,
\end{equation*}
and we find its more general solution as a restriction to $\mathbb{R}$ of an entire function of exponential type. As an important special case, we solve explicitly the linear nonlocal zeta field equation
\begin{equation*}
\zeta(\partial_t^2+h)ϕ= J(t)\; ,
\end{equation*}
in which $h$ is a real parameter, $\zeta$ is the Riemann zeta function, and $J$ is an entire function of exponential type. We also analyze the case in which $J$ is a more general analytic function (subject to some weak technical assumptions). This case turns out to be rather delicate: we need to re-interpret the symbol $\zeta(\partial_t^2+h)$ and to leave the class of functions of exponential type. We prove that in this case the zeta-nonlocal equation above admits an analytic solution on a Runge domain determined by $J$. The linear zeta field equation is a linear version of a field model depending on the Riemann zeta function arising from $p$-adic string theory."
[added 30th October 2019]
A. Dixit and A. Roy, "Analogue of a Fock-type integral arising from electromagnetism and its applications in number theory" (preprint 07/2019)
"Closed-form evaluations of certain integrals of $J_{0}(\xi)$, the Bessel
function of the first kind, have been crucial in the studies on the
electromagnetic field of alternating current in a circuit with two groundings, as can be seen from the works of Fock and Bursian, Schermann etc. Koshliakov's generalization of one such integral, which contains $J_s(\xi)$ in the integrand, encompasses several important integrals in the literature including Sonine's integral. Here we derive an analogous integral identity where $J_{s}(\xi)$ is replaced by a kernel consisting of a combination of
$J_{s}(\xi)$, $K_{s}(\xi)$ and $Y_{s}(\xi)$ that is of utmost importance in
number theory. Using this identity and the Voronoi summation formula, we derive a general transformation relating infinite series of products of Bessel functions $I_{\lambda}(\xi)$ and $K_{\lambda}(\xi)$ with those involving the Gaussian hypergeometric function. As applications of this transformation, several important results are derived, including what we believe to be a corrected version of the first identity found on page $336$ of Ramanujan's Lost Notebook."
[added 30th October 2019]
G. Lang, "Conjectures about the structure of strong- and weak-coupling expansions of a few ground-state observables in the Lieb–Liniger and Yang–Gaudin models" (preprint 07/2019)
[abstract:] "In this paper, we apply experimental number theory to two integrable quantum models in one dimension, the Lieb–Liniger Bose gas and the Yang–Gaudin Fermi gas with contact interactions. We identify patterns in weak- and strong-coupling series expansions of the ground-state energy, local correlation functions and pressure. Based on the most accurate data available in the literature, we make a few conjectures about their mathematical structure and extrapolate to higher orders."
[added 30th October 2019]
K. Blackwell, N. Borade, C. Devlin VI, N. Luntzlara, R. Ma, S.J. Miller, M. Wang and W. Xu, "Distribution of eigenvalues of random real symmetric block matrices" (preprint 08/2019)
"Random Matrix Theory (RMT) has successfully modeled diverse systems, from energy levels of heavy nuclei to zeros of $L$-functions. Many statistics in one can be interpreted in terms of quantities of the other; for example, zeros of $L$-functions correspond to eigenvalues of matrices, and values of $L$-functions to values of the characteristic polynomials. This correspondence has allowed RMT to successfully predict many number theory behaviors; however, there are some operations which to date have no RMT analogue. The motivation of this paper is to try and find an RMT equivalent to Rankin-Selberg convolution, which builds a new $L$-functions from an input pair.
For definiteness we concentrate on two specific families, the ensemble of palindromic real symmetric Toeplitz (PST) matrices and the ensemble of real symmetric (RS) matrices, whose limiting spectral measures are the Gaussian and semicircle distributions, respectively; these were chosen as they are the two extreme cases in terms of moment calculations. For a PST matrix $A$ and a RS matrix $B$, we construct an ensemble of random real symmetric block matrices whose first row is $\{A,B\}$ and whose second row is $\{B,A\}$. By Markov's Method of Moments, we show this ensemble converges weakly and almost surely to a new, universal distribution with a hybrid of Gaussian and semicircle behaviors. We extend this construction by considering an iterated concatenation of matrices from an arbitrary pair of random real symmetric sub-ensembles with different limiting spectral measures. We prove that finite iterations converge to new, universal distributions with hybrid behavior, and that infinite iterations converge to the limiting spectral measures of the component matrices."
[added 30th October 2019]
A.R. Fazely, "Prime-index parametrization for total neutrino-nucleon cross sections and $pp$ cross sections" (preprint 08/19)
"A prime number based parametrization for total neutrino-nucleon cross section
is presented. The method employs the relation between prime numbers and their
indices to reproduce neutrino cross sections for neutrino energies from the
$MeV$ to the $PeV$ regions where experimental data are available. This
prime-index relation provides estimates of the neutrino-nucleon cross sections valid across many decades of neutrino energy scales. The $PeV$ data are from the recently published astrophysical $\nu_{\mu} +\bar{\nu_{\mu}}$ rates in the IceCube detector as well as neutrino-nucleon cross section measurements. A similar method has been employed for high energy $pp$ cross sections which explains the $(\ln s)^2$ parametrization first proposed by Heisenberg."
[added 30th October 2019]
O. Fuentealba, H.A. González, M. Pino and R. Troncoso, "The anisotropic chiral boson" (preprint 09/19)
[abstract:] "We construct the theory of a chiral boson with anisotropic scaling, characterized by a dynamical exponent $z$, whose action reduces to that of Floreanini and Jackiw in the isotropic case ($z=1$). The standard free boson with Lifshitz scaling is recovered when both chiralities are nonlocally combined. Its canonical structure and symmetries are also analyzed. As in the isotropic case, the theory is also endowed with a current algebra. Noteworthy, the standard conformal symmetry is shown to be still present, but realized in a nonlocal way. The exact form of the partition function at finite temperature is obtained from the path integral, as well as from the trace over $\hat{u}(1)$ descendants. It is essentially given by the generating function of the number of partitions of an integer into $z$-th powers, being a well-known object in number theory. Thus, the asymptotic growth of the number of states at fixed energy, including subleading corrections, can be obtained from the appropriate extension of the renowned result of Hardy and Ramanujan."
[added 30th October 2019]
J. Ossorio-Castillo and J.M. Tornero, "An adiabatic quantum algorithm for the Frobenius problem" (preprint 07/2019)
[abstract:] "The (Diophantine) Frobenius problem is a well-known NP-hard problem (also called the stamp problem or the chicken nugget problem) whose origins lie in the realm of combinatorial number theory. In this paper we present an adiabatic quantum algorithm which solves it, using the so-called Apéry set of a numerical semigroup, via a translation into a QUBO problem. The algorithm has been specifically designed to run in a D-Wave 2X machine."
[added 7th July 2019]
Yu.I. Bogdanov, N.A. Bogdanova, D.V. Fastovets and V.F. Lukichev, "Representation of Boolean functions in terms of quantum computation" (preprint 06/2019)
[abstract:] "The relationship between quantum physics and discrete mathematics is reviewed in this article. The Boolean functions unitary representation is considered. The relationship between Zhegalkin polynomial, which defines the algebraic normal form of Boolean function, and quantum logic circuits is described. It is shown that quantum information approach provides simple algorithm to construct Zhegalkin polynomial using truth table. Developed methods and algorithms have arbitrary Boolean function generalization with multibit input and multibit output. Such generalization allows us to use many-valued logic ($k$-valued logic, where $k$ is a prime number). Developed methods and algorithms can significantly improve quantum technology realization. The presented approach is the baseline for transition from classical machine logic to quantum hardware."
[added 7th July 2019]
S. Matsutani, "A novel discrete theory of a screw dislocation in the BCC crystal lattice" (preprint 06/2019)
[abstract:] "In this paper, we proposed a novel method using the elementary number theory to investigate the discrete nature of the screw dislocations in crystal lattices, simple cubic (SC) lattice and body centered cubic (BCC) lattice, by developing the algebraic description of the dislocations in the previous report (Hamada, Matsutani, Nakagawa, Saeki, Uesaka, *Pacific J. Math. for Industry* **10** (2018), 3). Using the method, we showed that the stress energy of the screw dislocations in the BCC lattice and the SC lattice are naturally described; the energy of the BCC lattice was expressed by the truncated Epstein–Hurwitz zeta function of the Eisenstein integers, whereas that of SC lattice is associated with the truncated Epstein–Hurwitz zeta function of the Gauss integers."
[added 7th July 2019]
I. Huet, M. Rausch de Traubenberg and C. Schubert, "Dihedral invariant polynomials in the effective Lagrangian of QED" (preprint 06/2019)
[abstract:] "We present a new group-theoretical technique to calculate weak field expansions for some Feynman diagrams using invariant polynomials of the dihedral group. In particular we show results obtained for the first coefficients of the three loop effective Lagrangian of 1+1 QED in an external constant field, where the dihedral symmetry appears. Our results suggest that a closed form involving rational numbers and the Riemann zeta function might exist for these coefficients."
[added 7th July 2019]
F. Pausinger, "Greedy energy minimization can count in binary: point charges and the van der Corput sequence" (preprint 06/2019)
[abstract:] "This paper establishes a connection between a problem in potential theory and mathematical physics, arranging points so as to minimize an energy functional, and a problem in combinatorics and number theory, constructing ``well-distributed'' sequences of points on $[0,1]$. Let $f:[0,1] \rightarrow \mathbb{R}$ be (i) symmetric $f(x) = f(-x)$, (ii) twice differentiable on
$[0,1]$, and (iii) such that $f''(x)>0$ for all $x \in [0,1]$. We study the greedy dynamical system, where, given an initial set $\{x_0, \ldots, x_{N-1}\} \subset [0,1]$, the point $x_N$ is obtained as $$ x_{N} = \arg\min_x \sum_{k=0}^{N-1}{f(|x-x_k|)}.$$ We prove that if we start this construction
with a single element $x_0 \in [0,1]$, then all arising constructions are permutations of the van der Corput sequence (counting in binary and reflected about the comma): greedy energy minimization recovers the way we count in binary. This gives a new construction of the classical van der Corput sequence.
Interestingly, the point sets we derive are also known in a different context as Leja sequences on the unit disk. Finally, the special case $f(x) = 1-\log(2 \sin(\pi x))$ answers a question of Steinerberger."
[added 7th July 2019]
M. Lesiuk and B. Jeziorski, "Complete basis set extrapolation of electronic correlation energies using the Riemann zeta function" (preprint 05/2019)
[abstract:] "In this communication we present a method of complete basis set (CBS) extrapolation of correlation energies obtained with a systematic sequence of one-electron basis sets. Instead of fitting the finite-basis results with a certain functional form, we perform analytic re-summation of the missing contributions coming from higher angular momenta, $l$. The assumption that they vanish asymptotically as an inverse power of $l$ leads to an expression for the CBS limit given in terms of the Riemann zeta function. This result is turned into an extrapolation method that is very easy to use and requires no ``empirical'' parameters to be optimized. The performance of the proposed method is assessed by comparing the results with accurate reference data obtained with explicitly correlated theories and with results obtained with standard extrapolation schemes. On average, the errors of the zeta-function extrapolation are several times smaller compared with the conventional schemes employing the same number of points. A recipe for estimation of the residual extrapolation error is also proposed."
[added 7th July 2019]
A. Dabholkar, Ramanujan and quantum black holes" (preprint 05/2019)
[abstract:] "Explorations of quantum black holes in string theory have led to fascinating connections with the work of Ramanujan on partitions and mock theta functions, which in turn relate to diverse topics in number theory and enumerative geometry. This article aims to explain the physical significance of these interconnections."
[added 7th July 2019]
D. Li, "Entanglement classification via integer partitions" (preprint 05/2019)
[abstract:] "In [M. Walter et al., *Science* 340, 1205, 7 June (2013)], they gave a sufficient condition for genuinely entangled pure states and discussed SLOCC classification via polytopes and the eigenvalues of the single-particle states. In this paper, for $4n$ qubits, we show the invariance of algebraic multiplicities (AMs) and geometric multiplicities (GMs) of eigenvalues and the invariance of sizes of Jordan blocks (JBs) of the coefficient matrices under SLOCC. We explore properties of spectra, eigenvectors, generalized eigenvectors, standard Jordan normal forms (SJNFs), and Jordan chains of the coefficient matrices. The properties and invariance permit a reduction of SLOCC classification of $4n$ qubits to integer partitions (in number theory) of the number $2^{2n}-k$ and the AMs."
[added 7th July 2019]
D. Delmastro and J. Gomis, "Symmetries of Abelian Chern–Simons theories and arithmetic" (preprint 04/2019)
[abstract:] "We determine the unitary and anti-unitary Lagrangian and quantum symmetries of arbitrary abelian Chern–Simons theories. The symmetries depend sensitively on the arithmetic properties (e.g. prime factorization) of the matrix of Chern–Simons levels, revealing interesting connections with number theory. We give a complete characterization of the symmetries of abelian topological field theories and along the way find many theories that are non-trivially time-reversal invariant by virtue of a quantum symmetry, including $U(1)_k$ Chern–Simons theory and $(\mathbb{Z}_k)_l$ gauge theories. For example, we prove that $U(1)_k$ Chern–Simons theory is time-reversal invariant if and only if $âˆ’1$ is a quadratic residue modulo $k$, which happens if and only if all the prime factors of $k$ are Pythagorean (i.e., of the form $4n+1$), or Pythagorean with a single additional factor of $2$. Many distinct non-abelian finite symmetry groups are found.
[added 7th July 2019]
L. Vinet and H. Zhan, "Perfect state transfer on weighted graphs of the Johnson scheme" (preprint 04/2019)
[abstract:] "We characterize perfect state transfer on real-weighted graphs of the Johnson scheme $\mathcal{J}(n,k)$. Given $\mathcal{J}(n,k)=\{A_1, A_2, \cdots, A_k\}$
and $A(X) = w_0A_0 + \cdots + w_m A_m$, we show, using classical number theory
results, that $X$ has perfect state transfer at time $\tau$ if and only if
$n=2k$, $m\ge 2^{\lfloor{\log_2(k)} \rfloor}$, and there are integers $c_1,
c_2, \cdots, c_m$ such that (i) $c_j$ is odd if and only if $j$ is a power of
$2$, and (ii) for $r=1,2,\cdots,m$, \[w_r = \frac{\pi}{\tau} \sum_{j=r}^m
\frac{c_j}{\binom{2j}{j}} \binom{k-r}{j-r}.\] We then characterize perfect
state transfer on unweighted graphs of $\mathcal{J}(n,k)$. In particular, we
obtain a simple construction that generates all graphs of $\mathcal{J}(n,k)$
with perfect state transfer at time $\pi/2$."
[added 7th July 2019]
R. He, M.-Z. Ai, J.-M. Cui, Y.-F. Huang, Y.-J. Han, C.-F. Li and G.-C. Guo, "Finding the Riemann zeros by periodically driving a single trapped ion" (preprint 03/2019)
[abstract:] "The Riemann hypothesis implies the most profound secret of the prime numbers. It is still an open problem despite various attempts have been made by numerous mathematicians. One of the most fantastic approaches to treat this problem is to connect this hypothesis with the spectrum of a physical Hamiltonian. However, designing and performing a suitable Hamiltonian corresponding to this conjecture is always a primary challenge. Here we report the first experiment to find the non-trivial zeros of the Riemann function and Pólya's function using the novel approach proposed by Floquet method. In this approach, the zeros of the functions instead are characterized by the occurance of the crossings of the quasienergies when the dynamics of the system is frozen. With the properly designed periodically driving functions, we can experimentally obtain the first non-trivial zero of the Riemann function and the first two non-trivial zeros of Pólya's function which are in excellent agreement with their exact values. Our work provides a new insight for the Pólya–Hilbert
conjecture in quantum systems."
[added 23rd April 2019]
M.V.N. Murthy, M. Brack and R.K. Bhaduri, "On the asymptotic distinct prime partitions of integers" (preprint 04/2019)
[abstract:] "We discuss $Q(n)$, the number of ways a given integer $n$ may be written as a sum of distinct primes, and study its asymptotic form $Q_{as}(n)$ valid in the limit $n\to\infty$. We obtain $Q_{as}(n)$ by Laplace inverting the fermionic partition function of primes, in number theory called the generating function of the distinct prime partitions, in the saddle-point approximation. We find that our result of $Q_{as}(n)$, which includes two higher-order corrections to the leading term in its exponent and a pre-exponential correction factor, approximates the exact $Q(n)$ far better than its simple leading-order exponential form given so far in the literature."
[added 23rd April 2019]
R. Dong and M. Khalkhali, "Second quantization and the spectral action" (preprint 03/2019)
[abstract:] "We show that by incorporating chemical potentials one can extend the formalism of spectral action principle to bosonic second quantization. In fact we show that the von Neumann entropy, the average energy, and the negative free energy of the state defined by the bosonic, or fermionic, grand partition function can be expressed as spectral actions, and all spectral action coefficients can be given in terms of the modified Bessel functions. In the Fermionic case, we show that the spectral coefficients for the von Neumann entropy, in the limit when the chemical potential $\mu$ approaches to $0$, can be expressed in terms of the Riemann zeta function. This recovers a recent result of Chamseddine–Connes–van Suijlekom."
[added 23rd April 2019]
F. Gleisberg, F. Di Pumpo, G. Wolff and W.P. Schleich, "Prime factorization of arbitrary integers with a logarithmic energy spectrum" (preprint 02/2019)
[abstract:] "We propose an iterative scheme to factor numbers based on the quantum dynamics of an ensemble of interacting bosonic atoms stored in a trap where the single-particle energy spectrum depends logarithmically on the quantum number. When excited by a time-dependent interaction these atoms perform Rabi oscillations between the ground state and an energy state characteristic of the factors. The number to be factored is encoded into the frequency of the sinusoidally modulated interaction. We show that a measurement of the energy of the atoms at a time chosen at random yields the factors with probability one half. We conclude by discussing a protocol to obtain the desired prime factors employing a logarithmic energy spectrum which consists of prime numbers only."
[added 19th February 2019]
D. Momeni, "Bose–Einstein condensation for an exponential density of states function and Lerch zeta function" (preprint 02/2019)
[abstract:] "I showed that how Bose–Einstein condensation (BEC) in a non interacting bosonic system with exponential density of the states function yields to a new class of Lerch zeta functions. By looking on the critical temperature, I suggeted a possible strategy to prove the ``Riemann hypothesis'' problem. In a theorem and a lemma I suggested that the classical limit $\hbar\to 0$ of BEC can be used as a tool to find zeros of real part of the Riemann zeta function with complex argument. It reduces the Riemann hypothesis to a softer form. Furthermore I proposed a pair of creation-annihilation operators for BEC phenomena. These set of creation-annihilation operators is defined on a complex Hilbert space. They build a set up to interpret this type of BEC as a creation-annihilation phenomena of the virtual hypothetical particle."
[added 19th February 2019]
C.M. Newman and W. Wu, "Constants of de Bruijn-Newman type in analytic number theory and statistical physics" (preprint 01/2019)
[abstract:] "One formulation in 1859 of the Riemann Hypothesis (RH) was that the Fourier transform $H_f(z)$ of $f$ for $z \in \mathbb{C}$ has only real zeros when
$f(t)$ is a specific function $\Phi(t)$. Pólya's 1920s approach to RH
extended $H_f$ to $H_{f,\lambda}$, the Fourier transform of $e^{\lambda t^2}
f(t)$. We review developments of this approach to RH and related ones in
statistical physics where $f(t)$ is replaced by a measure $d \rho(t)$.
Pólya's work together with 1950 and 1976 results of de Bruijn and Newman,
respectively, imply the existence of a finite constant $\Lambda_{DN} =
\Lambda_{DN}(\Phi)$ in $(-\infty, 1/2]$ such that $H_{\Phi,\lambda}$ has only
real zeros if and only if $\lambda\geq \Lambda_{DN}$; RH is then equivalent to
$\Lambda_{DN} \leq 0$. Recent developments include the Rodgers and Tao proof of
the 1976 conjecture that $\Lamda_{DN} \geq 0$ (that RH, if true, is only
barely so) and the Polymath 15 project improving the $1/2$ upper bound to about $0.22$. We also present examples of $\rho$'s with differing $H_{\rho,\lambda}$
and $\Lambda_{DN}(\rho)$ behaviors; some of these are new and based on a
recent weak convergence theorem of the authors."
[added 19th February 2019]
K.R. Willison, "An intracellular calcium frequency code model extended to the Riemann zeta function" (preprint 05/2019, submitted to arXiv.org)
[abstract:] "We have used the Nernst chemical potential treatment to couple the time domains of sodium and calcium ion channel opening and closing rates to the spatial domain of the diffusing waves of the travelling calcium ions inside single cells. The model is plausibly evolvable with respect to the origins of the molecular components and the scaling of the system from simple cells to neurons. The mixed chemical potentials are calculated by summing the concentrations or particle numbers of the two constituent ions which are pure numbers and thus dimensionless. Chemical potentials are true thermodynamic free Gibbs/Fermi energies and the forces acting on chemical flows are calculated from the natural logarithms of the particle numbers or their concentrations. The mixed chemical potential is converted to the time domain of an action potential by assuming that the injection of calcium ions accelerates depolarization in direct proportion to the amplitude of the total charge contribution of the calcium pulse. We assert that the natural logarithm of the real component ($\zeta_n$) of the imaginary term ($\zeta_n i$) of any Riemann zeta zero ($1/2+\zeta_n i$) corresponds to an instantaneous calcium potential ($Z_n$). In principle, in a physiologically plausible fashion, the first few thousand Riemann $\zeta$-zeros can be encoded on this chemical scale manifested as regulated step-changes in the amplitudes of naturally occurring calcium current transients. We show that pairs of $Z_n$ channels can form Dirac fences which encode the logarithmic spacings and summed amplitudes of any pair of Riemann zeros. Remarkably the beat frequencies of the pairings of the early frequency terms ($Z_n-Z_{n+1}$, $Z_n-Z_{n+2},\dots$) overlap the naturally occurring frequency modes ($\gamma,\delta,\theta$) in vertebrate brains. Action potential control of calcium transients is a process whereby neuronal systems construct precise step functions; actually Dirac distributions which also underpin the Riemann mathematics. The equation for the time domain in the biological model has a similar form to the Riemann zeta function on the half-plane and mimics analytical continuation on the complex plane. Once coupled to neurophysiological binding processes these transients may underpin calculation in eukaryotic nervous systems."
[added 7th February 2019]
F.I. Moxley III, "Decidability of the Riemann Hypothesis" (preprint 09/2018)
[abstract:] "The Hamiltonian of a quantum mechanical system has an affiliated spectrum, and in order for
this spectrum to be observable, the Hamiltonian should be Hermitian. Such quantum Riemann zeta
function analogies have led to the Bender–Brody–Müller (BBM) conjecture, which involves a nonHermitian Hamiltonian whose eigenvalues are the nontrivial zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. In this
case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator
is for integrable quantum systems. As such, herein we perform a symmetrization procedure of the
BBM Hamiltonian to obtain a Hermitian Hamiltonian using a similarity transformation, and provide
an analytical expression for the eigenvalues of the results using Green's functions. A Gelfand triplet
is then used to ensure that the eigenvalues are well defined. The holomorphicity of the resulting
eigenvalues is demonstrated, and it is shown that that the expectation value of the Hamiltonian
operator is also zero such that the nontrivial zeros of the Riemann zeta function are not observable,
i.e., the Riemann Hypothesis is not decidable. Moreover, a second quantization of the resulting
Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic
continuation of the Riemann zeta function is obtained. Finally, from the holomorphicity of the
eigenvalues it is shown that the real part of every nontrivial zero of the Riemann zeta function exists
at $\sigma = 1/2$, and a general solution is obtained by performing an invariant similarity transformation"
[added 20th January 2019]
F.I. Moxley III, "A Schrödinger equation for solving the Bender–Brody–Muller Conjecture" (AIP Conference Proceedings, 2017)
[abstract:] "The Hamiltonian of a quantum mechanical system has an affiliated spectrum. If this spectrum is the sequence of prime numbers, a connection between quantum mechanics and the nontrivial zeros of the Riemann zeta function can be made. In this case, the Riemann zeta function is analogous to chaotic quantum systems, as the harmonic oscillator is for integrable quantum systems. Such quantum Riemann zeta function analogies have led to the Bender–Brody–Muller (BBM) conjecture, which involves a non-Hermitian Hamiltonian that maps to the zeros of the Riemann zeta function. If the BBM Hamiltonian can be shown to be Hermitian, then the Riemann Hypothesis follows. As such, herein we perform a symmetrization procedure of the BBM Hamiltonian to obtain a unique Hermitian Hamiltonian that maps to the zeros of the analytic continuation of the Riemann zeta function, and discuss the eigenvalues of the results. Moreover, a second quantization of the resulting Schrödinger equation is performed, and a convergent solution for the nontrivial zeros of the analytic continuation of the Riemann zeta function is obtained. Finally, the Hilbert–Pólya conjecture is discussed, and it is heuristically shown that the real part of every nontrivial zero of the Riemann zeta function converges at $\sigma = 1/2$."
[added 20th January 2019]
N. Dattani, *Quadratization in discrete optimization and quantum mechanics* (open-source book, 01/2019)
[abstract:] "A book about turning high-degree optimization problems into quadratic optimization problems that maintain the same global minimum (ground state). This book explores quadratizations for pseudo-Boolean optimization, perturbative gadgets used in QMA completeness theorems, and also non-perturbative $k$-local to $2$-local transformations used for quantum mechanics, quantum annealing and universal adiabatic quantum computing... **Applications cited include** computer vision problems (e.g. image de-noising, un-blurring, etc.), **number theory (e.g. integer factoring)**, graph theory (e.g. Ramsey number determination), and quantum chemistry."
[added 20th January 2019]
A. Huang, B. Stoica and S.-T. Yau, "General relativity from $p$-adic strings" (preprint 01/2019)
[abstract:] "For an arbitrary prime number $p$, we propose an action for bosonic $p$-adic strings in curved target spacetime, and show that the vacuum Einstein equations of the target are a consequence of worldsheet scaling symmetry of the quantum $p$-adic strings, similar to the ordinary bosonic strings case. It turns out that certain $p$-adic automorphic forms are the plane wave modes of the bosonic fields on $p$-adic strings, and that the regularized normalization of these modes on the $p$-adic worldsheet presents peculiar features which reduce part of the computations to familiar setups in quantum field theory, while also exhibiting some new features that make loop diagrams much simpler. Assuming a certain product relation, we also observe that the adelic spectrum of the bosonic string corresponds to the nontrivial zeros of the Riemann zeta function."
[added 20th January 2019]
T.C. Petersen, M. Ceko, I. D. Svalbe, M.J. Morgan, A.I. Bishop and D.M. Paganin, "Simple wave-optical superpositions as prime number sieves" (preprint 12/2018)
[abstract:] "We encode the sequence of prime numbers into simple superpositions of identical waves, mimicking the archetypal prime number sieve of Eratosthenes. The primes are identified as zeros accompanied by phase singularities in a physically generated wave-field for integer valued momenta. Similarly, primes are encoded in the diffraction pattern from a simple single aperture and in the harmonics of a single vibrating resonator. Further, diffraction physics connections to number theory reveal how to encode all Gaussian primes, twin-primes, and how to construct wave fields with amplitudes equal to the divisor function at integer spatial frequencies. Remarkably, all of these basic diffraction phenomena reveal that the naturally irregular sequence of primes can arise from trivially ordered wave superpositions."
[added 20th January 2019]
F. Bouzeffour and M. Garayev, "Some aspects of number theory related to phase operators" (preprint 12/2018)
[abstract:] "We first extend the multiplicativity property of arithmetic functions to the setting of operators on the Fock space. Secondly, we use phase operators to get representation of some extended arithmetic functions by operators on the Hardy space. Finally, we show that radial limits to the boundary of the unit disc in the Hardy space is useful in order to go back to the classical arithmetic functions. Our approach can be understudied as a transition from the classical number theory to quantum setting."
[added 20th January 2019]
S.S. Avancini, R.L.S. Farias, W.R. Tavares, "Neutral meson properties in hot and magnetized quark matter: A new magnetic field independent regularization scheme applied to NJL-type model" (preprint 12/2018)
[abstract:] "A magnetic field independent regularization scheme (zMFIR) based on the Hurwitz–Riemann zeta function is introduced. The new technique is applied to the regularization of the mean-field thermodynamic potential and mass gap equation within the $SU(2)$ Nambu–Jona–Lasinio model in a hot and magnetized medium. The equivalence of the new and the standard MFIR scheme is demonstrated. The neutral meson pole mass is calculated in a hot and magnetized medium and the advantages of using the new regularization scheme are shown."
[added 20th January 2019]
A.S. Mischenko, "Maslov's concept of phase transition from Bose–Einstein to Fermi–Dirac distribution: Results of interdisciplinary workshop in MSU" (preprint 11/2018)
[abstract:] "At the end of 2017, an interdisciplinary scientific seminar was organized at Moscow University, devoted to the study and development of a new scientific concept created by V.P. Maslov, allowing you to take a fresh look at the statistics of Bose–Einstein and Fermi–Dirac ideal gases. This new point of view allows us to interpret the indicated statistics as particular cases of statistical properties in number theory, on the one hand, and to indicate the limits of phase transitions from Bose to Fermi distributions."
[added 20th January 2019]
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