##
recently archived material
[Items are added to the top of this list as they are archived elsewhere.]
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]
Number Theory and Dynamics, March 25–29, 2019, University of Cambridge, UK
[added 12th October 2018]
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."
[added 12th October 2018]
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."
[added 12th October 2018]
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."
[added 12th October 2018]
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 $\zeta(s)$."
[added 12th October 2018]
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."
[added 12th October 2018]
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."
[added 12th October 2018]
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."
[added 12th October 2018]
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."
[added 12th October 2018]
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.
J.D. Cook, "The Riemann Hypothesis, the fine structure constant and the Todd function"
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."
[added 24th September 2018]
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."
[added 24th September 2018]
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."
[added 24th September 2018]
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."
[added 24th September 2018]
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."
[added 24th September 2018]
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."
[added 24th September 2018]
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."
[added 24th September 2018]
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."
[added 24th September 2018]
H. Li, F. Li and S. Kanemits, *Number Theory and Its Applications II* (World Scientific, 2018)
[added 24th September 2018]
W. Dittrich, *Reassessing Riemann's paper on the number of primes less than a given magnitude* (Springer, 2018)
[added 24th September 2018]
H. Montgomery, A. Nikeghbali and M.T. Rassias, eds., *Exploring the Riemann Zeta function* (Springer 2017)
[added 24th September 2018]
**archive
mystery
guestbook
search
home
**
**
****
** |