##
recently archived material
[Items are added to the top of this list as they are archived elsewhere.]
"Number Theory, Strings, and Quantum Physics", an online conference, 1–5 June 2021
[added 23rd April 2021]
W. Bietenholz, "From Ramanujan to renormalization: The art of doing away with divergences and arriving at physical results" (preprint 02/2021)
[abstract:] "A century ago Srinivasa Ramanujan – the great self-taught Indian genius of mathematics – died, shortly after returning from Cambridge, UK, where he had collaborated with Godfrey Hardy. Ramanujan contributed numerous outstanding results to different branches of mathematics, like analysis and number theory, with a focus on special functions and series. Here we refer to apparently weird values which he assigned to two simple divergent series, $\sum_{n\geq 1} n$ and $\sum_{n\geq 1}n^3$. These values are sensible, however, as analytic continuations, which correspond to Riemann's $\zeta$-function. Moreover, they have applications in physics: we discuss the vacuum energy of the photon field, from which one can derive the Casimir force, which has been experimentally measured. We also discuss its interpretation, which remains controversial. This is a simple way to illustrate the concept of renormalization, which is vital in quantum field theory."
[added 23rd April 2021]
P. Dutta and D. Ghoshal, "Phase operator on $L_2(\mathbb{Q}_p$) and the zeroes of Fisher and Riemann" (preprint 02/2021)
[abstract:] "The distribution of the non-trivial zeroes of the Riemann zeta function, according to the Riemann hypothesis, is tantalisingly similar to the zeroes of the partition functions (Fisher and Yang–Lee zeroes) of statistical mechanical models studied by physicists. The resolvent function of an operator akin to the phase operator, conjugate to the number operator in quantum mechanics, turns out to be important in this approach. The generalised Vladimirov derivative acting on the space $L_2(\mathbb{Q}_p)$ of complex valued locally constant functions on the $p$-adic field is rather similar to the number operator. We show that a `phase operator' conjugate to it can be constructed on a subspace $L_2(p^{-1}\mathbb{Z}_p)$ of $L_2(\mathbb{Q}_p)$. We discuss (at physicists' level of rigour) how to combine this for all primes to possibly relate to the zeroes of the Riemann zeta function. Finally, we extend these results to the family of Dirichlet $L$-functions, using our recent construction of Vladimirov derivative like pseudodifferential operators associated with the Dirichlet characters."
[added 23rd April 2021]
A. T. DeCelles, "Global automorphic Sobolev theory and the automorphic heat kernel" (preprint 02/2021)
[abstract:] "Heat kernels arise in a variety of contexts including probability, geometry, and functional analysis; the automorphic heat kernel is particularly important in number theory and string theory. The typical construction of an automorphic heat kernel as a Poincaré series presents analytic difficulties, which can be dealt with in special cases (e.g. hyperbolic spaces) but are often sidestepped in higher rank by restricting to the compact quotient case. In this paper, we present a new approach, using global automorphic Sobolev theory, a robust framework for solving automorphic PDEs that does not require any simplifying assumptions about the rank of the symmetric space or the compactness of the arithmetic quotient. We construct an automorphic heat kernel via its automorphic spectral expansion in terms of cusp forms, Eisenstein series, and residues of Eisenstein series. We then prove uniqueness of the automorphic heat kernel as an application of operator semigroup theory. Finally, we prove the smoothness of the automorphic heat kernel by proving that its automorphic spectral expansion converges in the $C^{\infty}$-topology."
[added 23rd April 2021]
R. He, M.-Z. Ai, J.-M. Cui, Y.-F. Huang, Y.-J. Han, C.-F. Li, G.-C. Guo, G. Sierra and C.E. Creffield, "Riemann zeros from a periodically-driven trapped ion" (preprint 02/2021)
[abstract:] "The non-trivial zeros of the Riemann zeta function are central objects in number theory. In particular, they enable one to reproduce the prime numbers. They have also attracted the attention of physicists working in Random Matrix Theory and Quantum Chaos for decades. Here we present an experimental observation of the lowest non-trivial Riemann zeros by using a trapped ion qubit in a Paul trap, periodically driven with microwave fields. The waveform of the driving is engineered such that the dynamics of the ion is frozen when the driving parameters coincide with a zero of the real component of the zeta function. Scanning over the driving amplitude thus enables the locations of the Riemann zeros to be measured experimentally to a high degree of accuracy, providing a physical embodiment of these fascinating mathematical objects in the quantum realm."
[added 23rd April 2021]
G. Chavez and A. Allawala, "Prime zeta function statistics and Riemann zero-difference repulsion" (preprint 02/2021)
[abstract:] "We present a derivation of the numerical phenomenon that differences between the Riemann zeta function's nontrivial zeros tend to avoid being equal to the imaginary parts of the zeros themselves, a property called statistical ``repulsion'' between the zeros and their differences. Our derivation relies on the statistical properties of the prime zeta function, whose singularity structure specifies the positions of the Riemann zeros. We show that the prime zeta function on the critical line is asymptotically normally distributed with a covariance function that is closely approximated by the logarithm of the Riemann zeta function's magnitude on the $1$-line. This creates notable negative covariance at separations approximately equal to the imaginary parts of the Riemann zeros. This covariance function and the singularity structure of the prime zeta function combine to create a conditional statistical bias at the locations of the Riemann zeros that predicts the zero-difference repulsion effect. Our method readily generalizes to describe similar effects in the zeros of related Dirichlet $L$-functions."
[added 23rd April 2021]
G. Mussardo and A. LeClair, "Randomness of Mobius coefficents and brownian motion: Growth of the Mertens function and the Riemann Hypothesis" (preprint 01/2021)
[abstract:] "The validity of the Riemann Hypothesis (RH) on the location of the non-trivial zeros of the Riemann zeta-function is directly related to the growth of the Mertens function: the RH is indeed true if the Mertens function goes asymptotically as $M(x) \simeq x^{1/2+\epsilon}$. We show that this behavior can be established on the basis of a new probabilistic approach based on the global properties of the Mertens function. To this aim, we focus the attention on the square-free numbers and we derive a series of probabilistic results concerning the prime number distribution along the series of square-free numbers, the average number of prime divisors, the Erdos–Kac theorem for square-free numbers, etc. These results lead us to the conclusion that the Mertens function is subject to a normal distribution as much as any other random walk, therefore with an asymptotic behaviour given by $x^{1/2+\epsilon}$. We also argue how the Riemann Hypothesis implies the Generalised Riemann Hypothesis for the Dirichlet $L$-functions. Next we study the local properties of the Mertens function dictated by the Mobius coefficients restricted to the square-free numbers. We perform a massive statistical analysis on these coefficients, applying to them a series of randomness tests of increasing precision and complexity. The successful outputs of all these tests (with a level of confidence of 99% that all the sub-sequences analyzed are indeed random) can be seen as impressive experimental confirmations of the probabilistic normal law distribution of the Mertens function analytically established earlier. In view of the theoretical probabilistic argument and the large battery of statistical tests, we can conclude that while a violation of the RH is strictly speaking not impossible, it is however ridiculously improbable."
[added 23rd April 2021]
T. Kimura, "A refinement of Sato–Tate conjecture" (preprint 01/2021)
[abstract:] "We propose a refined version of the Sato–Tate conjecture about the spacing distribution of the angle determined for each prime number. We also discuss its implications on $L$-function associated with elliptic curves in the relation to random matrix theory."
[added 23rd April 2021]
T. Benoist, N. Cuneo, V. Jakšić and C.-A. Pillet, "On entropy production of repeated quantum measurements, II: Examples" (preprint 12/2020)
[abstract:] "We illustrate the mathematical theory of entropy production in repeated quantum measurement processes developed in a previous work by studying examples of quantum instruments displaying various interesting phenomena and singularities. We emphasize the role of the thermodynamic formalism, and give many examples of quantum instruments whose resulting probability measures on the space of infinite sequences of outcomes (shift space) do not have the (weak) Gibbs property. We also discuss physically relevant examples where the entropy production rate satisfies a large deviation principle but fails to obey the central limit theorem and the fluctuation-dissipation theorem. Throughout the analysis, we explore the connections with other, a priori unrelated topics like functions of Markov chains, hidden Markov models, matrix products **and number theory**."
[added 23rd April 2021]
I.O. Goriachuk and A.L. Kataev, "Riemann $\zeta(4)$ function contributions to $O(\alpha_s^5)$ terms of Adler D-function and Bjorken polarized sum rule in $SU(N_c)$ QCD: Results and consequences" (preprint 11/2020)
[abstract:] "Two renormalization group invariant quantities in quantum chromodinamics (QCD), defined in Euclidean space,namely, Adler D-function of electron-positron annihilation to hadrons and Bjorken polarized deep-inelastic scattering sum rule, are considered. It is shown, that the 5th order corrections to them in $\overline{MS}$-like renormalization prescriptions, proportional to Riemann $\zeta$-function $\zeta(4)$, can be restored by the transition to the C-scheme, with the $\beta$-function, analogous to Novikov, Shifman, Vainshtein and Zakharov exact $\beta$-function in $\mathcal{N} = 1$ supersymmetric gauge theories. The general analytical expression for these corrections in $SU(N_c)$ QCD is deduced and their scale invariance is shown. The $\beta$-expansion procedure for these contributions is performed and mutual cancellation of them in the 5th order of the generalized Crewther identity are discussed."
[added 23rd April 2021]
K. Shirish, "S-duality and chaos" (preprint 11/2020)
[abstract:] "The Renormalization group in field theories happens to resemble dynamical systems in many ways. In this paper, we discuss the unexpected connection between chaos and duality in field theories. In a sense, that various dual field theories can emerge at the end of chaotic RG trajectories, and hence strong-weak duality in quantum field theory is a direct result of the chaotic flow of the renormalization group. This suggests that various properties of field and string theories could come into existence due to chaotic RG flow. We also conjecture the existence of dual quantum field theories in the half strip of the Riemann zeta function."
[added 23rd April 2021]
A. Saxena, A. Shukla and A. Pathak, "A hybrid scheme for prime factorization and its experimental implementation using IBM quantum processor" (preprint 09/2020)
[abstract:] "We report a quantum-classical hybrid scheme for factorization of bi-prime numbers (which are odd and square-free) using IBM's quantum processors. The hybrid scheme proposed here involves both classical optimization techniques and adiabatic quantum optimization techniques, and is build by extending a previous scheme of hybrid factorization. The quantum part of the scheme is very general in the sense that it can be implemented using any quantum computing architecture. Here, as an example, we experimentally implement our scheme for prime factorization using IBM's QX4 quantum processor and have factorised $35$."
[added 23rd April 2021]
G. Caginalp and B. Ion, "Probabilistic renormalization and analytic continuation" (preprint 08/2020)
[abstract:] "We introduce a theory of probabilistic renormalization for series, the renormalized values being encoded in the expectation of a certain random variable on the set of natural numbers. We identify a large class of weakly renormalizable series of Dirichlet type, whose analysis depends on the properties of a (infinite order) difference operator that we call Bernoulli operator. For the series in this class, we show that the probabilistic renormalization is compatible with analytic continuation. The general zeta series for $s\neq 1$ is found to be strongly renormalizable and its renormalized value is given by the Riemann zeta function."
[added 23rd April 2021]
J.L. Rosales, S. Briongos and V. Martin, "Quantum chaos and the spectrum of factoring" (preprint 08/2020)
[abstract:] "There exists a Hamiltonian formulation of the factorisation problem which also needs the definition of a factorisation ensemble (a set to which factorable numbers, $N'=x'y'$, having the same trivial factorisation algorithmic complexity, belong). For the primes therein, a function $E$, that may take only discrete values, should be the analogous of the energy from a confined system of charges in a magnetic trap. This is the quantum factoring simulator hypothesis connecting quantum mechanics with number theory. In this work, we report numerical evidence of the existence of this kind of discrete spectrum from the statistical analysis of the values of $E$ in a sample of random OpenSSL $n$-bits moduli (which may be taken as a part of the factorisation ensemble). Here, we show that the unfolded distance probability of these $E$'s fits to a \textit{Gaussian Unitary Ensemble}, consistently as required, if they actually correspond to the quantum energy levels spacing of a magnetically confined system that exhibits chaos. The confirmation of these predictions bears out the quantum simulator hypothesis and, thereby, it points to the existence of a liaison between quantum mechanics and number theory. Shor's polynomial time complexity of the quantum factorisation problem, from pure quantum simulation primitives, was obtained."
[added 23rd April 2021]
E. Lupercio, "Non-commutative geometry indomitable" (preprint 08/2020)
[abstract:] "This paper is a very brief and gentle introduction to non-commutative geometry geared primarily towards physicists and geometers. It starts with a brief historical description of the motivation for non-commutative geometry and then goes on to motivate the subject from the point of view of the the understanding of local symmetries affordee by the theory of groupoids. The paper ends with a very rapid survey of recent developments and applications such as non-commutative toric geometry, the standard model for particle physics and the study of the Riemann Hypothesis."
[added 23rd April 2021]
R. Schneider, "Eulerian series, zeta functions and the arithmetic of partitions" (preprint 08/2020)
[abstract:] "In this Ph.D. dissertation (2018, Emory University) we prove theorems at the intersection of the additive and multiplicative branches of number theory, bringing together ideas from partition theory, $q$-series, algebra, modular forms and analytic number theory. We present a natural multiplicative theory of integer partitions (which are usually considered in terms of addition), and explore new classes of partition-theoretic zeta functions and Dirichlet series – as well as ``Eulerian'' $q$-hypergeometric series – enjoying many interesting relations. We find a number of theorems of classical number theory and analysis arise as particular cases of extremely general combinatorial structure laws. Among our applications, we prove explicit formulas for the coefficients of the $q$-bracket of Bloch–Okounkov, a partition-theoretic operator from statistical physics related to quasi-modular forms; we prove partition formulas for arithmetic densities of certain subsets of the integers, giving $q$-series formulas to evaluate the Riemann zeta function; we study $q$-hypergeometric series related to quantum modular forms and the "strange" function of Kontsevich; and we show how Ramanujan's odd-order mock theta functions (and, more generally, the universal mock theta function $g_3$ of Gordon–McIntosh) arise from the reciprocal of the Jacobi triple product via the $q$-bracket operator, connecting also to unimodal sequences in combinatorics and quantum modular-like phenomena."
[added 23rd April 2021]
B. Dragovich, "From $p$-adic to zeta strings" (preprint 07/2020)
[abstract:] "This article is related to construction of zeta strings from $p$-adic ones. In addition to investigation of $p$-adic string for a particular prime number $p$, it is also interesting to study collective effects taking into account all primes $p$. An idea behind this approach is that a zeta string is a whole thing with infinitely many faces which we see as $p$-adic strings. The name zeta string has origin in the Riemann zeta function contained in related Lagrangian. The starting point in construction a zeta string is Lagrangian for a $p$-adic open string. There are two types of approaches to get a Lagrangian for zeta string from Lagrangian for $p$-adic strings: additive and multiplicative approaches, that are related to two forms of the definition of the Riemann zeta function. As a result of differences in approaches, one obtains several different Lagrangians for zeta strings. We briefly discuss some properties of these Lagrangians, related potentials, equations of motion, mass spectrum and possible connection with ordinary strings. This is a review of published papers with some new views."
[added 23rd April 2021]
S. Dwivedi, V. Kumar Singh and A. Roy, "Semiclassical limit of topological Rényi entropy in $3d$ Chern–Simons theory" (preprint 07/2020)
[abstract:] "We study the multi-boundary entanglement structure of the state associated with the torus link complement $S^3\setminus T_{p,q}$ in the set-up of three-dimensional $SU(2)$_k Chern–Simons theory. The focal point of this work is the asymptotic behavior of the Rényi entropies, including the entanglement entropy, in the semiclassical limit of $k\to\infty$. We present a detailed analysis for several torus links and observe that the entropies converge to a finite value in the semiclassical limit. We further propose that the large $k$ limiting value of the Rényi entropy of torus links of type $T_{p,pn}$ is the sum of two parts: (i) the universal part which is independent of $n$, and (ii) the non-universal or the linking part which explicitly depends on the linking number $n$. Using the analytic techniques, we show that the universal part comprises of Riemann zeta functions and can be written in terms of the partition functions of two-dimensional topological Yang–Mills theory. More precisely, it is equal to the Rényi entropy of certain states prepared in topological $2d$ Yang–Mills theory with $SU(2)$ gauge group. Further, the universal parts appearing in the large $k$ limits of the entanglement entropy and the minimum Rényi entropy for torus links $T_{p,pn}$ can be interpreted in terms of the volume of the moduli space of flat connections on certain Riemann surfaces. We also analyze the Rényi entropies of $T_{p,pn}$ link in the double scaling limit of $k\to\infty$ and $n\to\infty$ and propose that the entropies converge in the double limit as well."
[added 23rd April 2021]
L.-P. Arguin, P. Bourgade and M. Radziwiłł, The Fyodorov–Hiary–Keating Conjecture, I" (preprint 07/2020)
[abstract:] "By analogy with conjectures for random matrices, Fyodorov–Hiary–Keating and Fyodorov–Keating proposed precise asymptotics for the maximum of the Riemann zeta function in a typical short interval on the critical line. In this paper, we settle the upper bound part of their conjecture in a strong form. More precisely, we show that the measure of those $T\leq t \leq 2T$ for which
\[ \max_{|h|\leq 1}|\zeta(1/2 + it + ih)| > e^y \frac{\log T}{(\log \log T)^{3/4}} \]
is bounded by $Cye^{-2y} uniformly in $y\geq 1$. This is expected to be optimal for $y = O(\sqrt{\log\logT})$. This upper bound is sharper than what is known in the context of random matrices, since it gives (uniform) decay rates in $y$. In a subsequent paper we will obtain matching lower bounds."
[added 23rd April 2021]
D. Lebiedz, "Holomorphic Hamiltonian $\xi$-flow and Riemann zeros"
[abstract:] "With a view on the formal analogy between the Riemann–von Mangoldt explicit formula and semiclassical quantum mechanics in terms of the Gutzwiller trace formula we construct a complex-valued Hamiltonian $H(q,p)=\xi(q)p$ from the holomorphic flow $q˙=\xi(q)$ and its variational differential equation. The Hamiltonian phase portrait $q(p)$ is a Riemann surface equivalent to reparameterized $\xi$-Newton flow solutions in complex-time, its flow map differential is determined by all Riemann zeros and reminiscent of a `spectral sum' in trace formulas. Canonical quantization for particle quantum mechanics on a circle leads to a Dirac-type momentum operator with discrete spectrum given by classical closed orbit periods determined by derivatives $\xi'(\rho_n)$ at Riemann zeros."
[added 23rd April 2021]
S. Ouvry and A. Polychronakos, "Lattice walk area combinatorics, some remarkable trigonometric sums and Apéry-like numbers"
[abstract:] "Explicit algebraic area enumeration formulae are derived for various lattice walks generalizing the canonical square lattice walk, and in particular for the triangular lattice chiral walk recently introduced by the authors. A key element in the enumeration is the derivation of some remarkable identities involving trigonometric sums – which are also important building blocks of non trivial quantum models such as the Hofstadter model – and their explicit rewriting in terms of multiple binomial sums. An intriguing connection is also made with number theory and some classes of Apéry-like numbers, the cousins of the Ap–ry numbers which play a central role in irrationality considerations for $\zeta(2)$ and $\zeta(3)$."
[added 23rd April 2021]
E.C. Bailey and J.P. Keating, "On the moments of the moments of $\zeta(1/2+it)$" (preprint 06/2020)
[abstract:] "Taking $t$ at random, uniformly from $[0,T]$, we consider the $k$th moment, with respect to $t$, of the random variable corresponding to the $2\beta$th moment of $\zeta(1/2+ix)$ over the interval $x\in (t,t+1]$, where $\zeta(s)$ is the Riemann zeta function. We call these the `moments of moments' of the Riemann zeta function, and present a conjecture for their asymptotics, when $T\to\infty$, for integer $k$, $\beta$. This is motivated by comparisons with results for the moments of moments of the characteristic polynomials of random unitary matrices and is shown to follow from a conjecture for the shifted moments of $\zeta(s)$ due to Conrey, Farmer, Keating, Rubinstein, and Snaith. Specifically, we prove that a function which, the shifted-moment conjecture of implies, is a close approximation to the moments of moments of the zeta function does satisfy the asymptotic formula that we conjecture. We motivate as well similar conjectures for the moments of moments for other families of primitive $L$-functions."
[added 23rd April 2021]
W.-J. Rao, "Higher-order level spacings in random matrix theory based on Wigner's conjecture" (preprint 05/2020)
[abstract:] "The distribution of higher order level spacings, i.e. the distribution of $\{s^{(n)}_i = E_{i+n} − E_i\}$ with $n\geq 1$ is derived analytically using a Wigner-like surmise for Gaussian ensembles of random matrix as well as Poisson ensemble. It is found $s^{(n)}$ in Gaussian ensembles follows a generalized Wigner–Dyson distribution with rescaled parameter $\alpha = \nu C^2_{n+1} + n − 1$, while that in Poisson ensemble follows a generalized semi-Poisson distribution with index $n$. Numerical evidences are provided through simulations of random spin systems as well as non-trivial zeros of Riemann zeta function. The higher order generalizations of gap ratios are also discussed."
[added 23rd April 2021]
Y. Nellambakam and K.V.S. Shiv Chaitanya, "Negative refractive index, perfect lens and Cesàro convergence" (preprint 05/2020)
[abstract:] "In this letter, we show that the restoration of evanescent wave in perfect lens obeys a new kind of convergence known as Cesaro convergence. Cesaro convergence allows us to extend the domain of convergence that is analytically continuing to the complex plane in terms of Riemann zeta function. Therefore, from the properties of Riemann zeta function we show that it is not possible to restore the evanescent wave for all the values of $r'_z$, [here $r'_z$ is complex]. The special value, that is, $r'_z = 1 = 2+ib$ refers to the non-existence of evanescent wave, is the physicists proof of Riemann Hypothesis."
[added 23rd April 2021]
M. Kovačević, "Signaling to relativistic observers: An Einstein–Shannon–Riemann encounter" (preprint 05/2020)
[abstract:] "A communication scenario is described involving a series of events triggered by a transmitter and observed by a receiver experiencing relativistic time dilation. The message selected by the transmitter is assumed to be encoded in the events' timings and is required to be perfectly recovered by the receiver, regardless of the difference in clock rates in the two frames of reference. It is shown that the largest proportion of the space of all $k$-event signals that can be selected as a code ensuring error-free information transfer in this setting equals $\zeta(k)^{-1}$, where $\zeta$ is the Riemann zeta function."
[added 23rd April 2021]
D. García-Martín, E. Ribas, S. Carrazza, J.I. Latorre and G. Sierra, "The prime state and its quantum relatives" (preprint 05/2020)
[abstract:] "The prime state of $n$ qubits, $|\mathbb{P}_n\rangle$, is defined as the uniform superposition of all the computational-basis states corresponding to prime numbers smaller than $2^n$. This state encodes, quantum mechanically, arithmetic properties of the primes. We first show that the quantum Fourier transform of the prime state provides a direct access to Chebyshev-like biases in the distribution of prime numbers. We next study the entanglement entropy of $n$ qubits, $|\mathbb{P}_n\rangle$ up to $n = 30$ qubits, and find a relation between its scaling and the Shannon entropy of the density of square-free integers. This relation also holds when the prime state is constructed using a qudit basis, showing that this property is intrinsic to the distribution of primes. The same feature is found when considering states built from the superposition of primes in arithmetic progressions. Finally, we explore the properties of other number-theoretical quantum states, such as those defined from odd composite numbers, square-free integers and starry primes. For this study, we have developed an open-source library that diagonalizes matrices using floats of arbitrary precision."
[added 22nd June 2020]
G. Mussardo, A. Trombettoni and Z. Zhang, "Prime suspects in a quantum ladder" (preprint 05/2020)
[abstract:] "In this paper we set up a suggestive number theory interpretation of a quantum ladder system made of $\mathcal{N}$ coupled chains of spin $1/2$. Using the hard-core boson representation, we associate to the spins $\sigma_a$ along the chains the prime numbers $p_a$ so that the chains become quantum registers for square-free integers. The Hamiltonian of the system consists of a hopping term and a magnetic field along the chains, together with a repulsion rung interaction and a permutation term between
next neighborhood chains . The system has various phases, among which there is one whose ground state is a coherent superposition of the first $\mathcal{N}$ prime numbers. We also discuss the realization of such a model in terms of an open quantum system with a dissipative Lindblad dynamics."
[added 22nd June 2020]
A. Saldivar, N.F. Svaiter and C.A.D. Zarro, "Functional equations for regularized zeta-functions and diffusion processes" (preprint 04/2020)
[abstract:] "We discuss modifications in the integral representation of the Riemann zeta-function that lead to generalizations of the Riemann functional equation that preserves the symmetry $s\to (1-s)$ in the critical strip. By modifying one integral representation of the zeta-function with a cut-off that does exhibit the symmetry $x\mapsto 1/x$, we obtain a generalized functional equation involving Bessel functions of second kind. Next, with another cut-off that does exhibit the same symmetry, we obtain a generalization for the functional equation involving only one Bessel function of second kind. Some connection between one regularized zeta-function and the Laplace transform of the heat kernel for the Euclidean and hyperbolic space is discussed."
[added 22nd June 2020]
L.A. Takhtajan, "Etudes of the resolvent" (preprint 04/2020)
[abstract:] "Based on the notion of the resolvent and on the Hilbert identities, this paper presents a number of classical results in the theory of differential operators and some of their applications to the theory of automorphic functions and number theory from a unified point of view. For instance, for the Sturm–Liouville operator there is a derivation of the Gelfand–Levitan trace formula, and for the one-dimensional Schroedinger operator a derivation of Faddeev's formula for the characteristic determinant and the Zakharov–Faddeev trace identities. Recent results on the spectral theory of a certain functional-difference operator arising in conformal field theory are then presented. The last section of the survey is devoted to the Laplace operator on a fundamental domain of a Fuchsian group of the first kind on the Lobachevsky plane. An algebraic scheme is given for proving analytic continuation of the integral kernel of the resolvent of the Laplace operator and the Eisenstein–Maass series. In conclusion, there is a discussion of the relation between the values of the Eisenstein–Maass series at Heegner points and Dedekind zeta-functions of imaginary quadratic fields, and it is explained why pseudo-cuspforms for the case of the modular group do not provide any information about the zeros of the Riemann zeta-function."
[added 22nd June 2020]
P. Betzios, N. Gaddam and O. Papadoulaki, "Black holes, quantum chaos, and the Riemann hypothesis" (preprint 04/2020)
[abstract:] "Quantum gravity is expected to gauge all global symmetries of effective theories, in the ultraviolet. Inspired by this expectation, we explore the consequences of gauging CPT as a quantum boundary condition in phase space. We find that it provides for a natural semiclassical regularisation and discretisation of the continuous spectrum of a quantum Hamiltonian related to the Dilation operator. We observe that the said spectrum is in correspondence with the zeros of the Riemann zeta and Dirichlet beta functions. Following ideas of Berry and Keating, this may help the pursuit of the Riemann hypothesis. It strengthens the proposal that this quantum Hamiltonian captures the dynamics of the scattering matrix on a Schwarzschild black hole background, given the rich chaotic spectrum upon discretisation. It also explains why the spectrum appears to be erratic despite the unitarity of the scattering matrix."
[added 22nd June 2020]
I. Bengtsson, "SICs: Some explanations" (preprint 04/2020)
[abstract:] "The problem of constructing maximal equiangular tight frames or SICs was raised by Zauner in 1998. Four years ago it was realized that the problem is closely connected to a major open problem in number theory. We discuss why such a connection was perhaps to be expected, and give a simplified sketch of some developments that have taken place in the past four years. The aim, so far unfulfilled, is to prove existence of SICs in an infinite sequence of dimensions."
[added 22nd June 2020]
B. Mukhametzhanov and S. Pal, "Beurling–Selberg extremization and modular bootstrap at high energies" (preprint 03/2020)
[abstract:] "We consider previously derived upper and lower bounds on the number of operators in a window of scaling dimensions $[\Delta-\delta,\Delta+\delta]$ at asymptotically large $\Delta$ in $2d$ unitary modular invariant CFTs. These bounds depend on a choice of functions that majorize and minorize the characteristic function of the interval $[\Delta-\delta,\Delta+\delta]$ and have Fourier transforms of finite support. The optimization of the bounds over this choice turns out to be exactly the Beurling–Selberg extremization problem, widely known in analytic number theory. We review solutions of this problem and present the corresponding bounds on the number of operators for any $\delta\geq 0$. When $2\delta \in \mathbb{Z}_{\geq 0}$ the bounds are saturated by known partition functions with integer-spaced spectra. Similar results apply to operators of fixed spin and Virasoro primaries in $c > 1$ theories."
[added 22nd June 2020]
M. Nardelli and A. Narelli, "On some Ramanujan formulas: Mathematical connections with $\phi$ and several parameters of quantum geometry of space, string theory and particle physics ($f_0(1710)$ scalar meson)" (preprint 04/2020)
M. Nardelli and A. Narelli, "On some Ramanujan formulas: mathematical connections with $\phi$ and several parameters of quantum geometry of space, string theory and particle physics, II" (preprint 04/2020)
[abstract:] "In this paper we have described and analyzed some Ramanujan expressions. We have obtained several mathematical connections with $\phi$ and various parameters of quantum geometry of space, string theory and particle physics."
[added 22nd June 2020]
P. Dutta and D. Ghoshal, "Pseudodifferential operators on $\mathbf{Q}_p$ and $L$-series" (preprint 02/2020)
[abstract:] "We define a family of pseudodifferential operators on the Hilbert space
$L^2(\mathbf{Q}_p)$ of complex valued square-integrable functions on the $p$-adic number field $\mathbf{Q}_p$. The Riemann zeta-function and the related Dirichlet $L$-functions can be expressed as a trace of these operators on a subspace of $L^2(\mathbf{Q}_p)$. We also extend this to the $L$-functions associated with modular (cusp) forms. Wavelets on $L^2(\mathbf{Q}_p)$ are common sets of eigenfunctions of these operators."
[added 12th March 2020]
S. Tafazoli, "Divergent integrals, the Riemann Zeta function, and the vacuum" (preprint 02/2020)
[abstract:] "This paper presents a new estimate for the vacuum energy density by summing the contributions of all quantum fields vacuum states which turns out to be in the same order of magnitude (but with opposite sign) as the predictions of current cosmological models and all observational data to date. The basis for this estimate is the recent results on the analytical solution to improper integral of divergent power functions using the Riemann Zeta function."
[added 12th March 2020]
M. McGuigan, "Riemann hypothesis, modified Morse potential and supersymmetric quantum mechanics" (preprint 02/2020)
[abstract:] "In this paper we discuss various potentials related to the Riemann zeta function and the Riemann Xi function. These potentials are modified versions of Morse potentials and can also be related to modified forms of the radial harmonic oscillator and modified Coulomb potential. We use supersymmetric quantum mechanics to construct their ground state wave functions and the Fourier transform of the ground state to exhibit the Riemann zeros. This allows us to formulate the Riemann hypothesis in terms of the location of the nodes of the ground state wave function in momentum space. We also discuss the relation these potentials to one and two matrix integrals and construct a few orthogonal polynomials associated with the matrix models. We relate the Schr\"odinger equation in momentum space to and finite difference equation in momentum space with an infinite number of terms. We computed the uncertainty relations associated with these potentials and ground states as well as the Shannon Information entropy and compare with the unmodified Morse and harmonic oscillator potentials. Finally we discuss the extension of these methods to other functions defined by a Dirichlet series such as the the Ramanujan zeta function."
[added 12th March 2020]
S. Tyagi, "Evaluation of exponential sums and Riemann zeta function on quantum computer" (preprint 02/2020)
[abstract:] "We show that exponential sums (ES) of the form
\begin{equation*}
s(f,N)= \sum_{k=0}^{N-1} \sqrt{w_k} e^{2 \pi i f(k)},
\end{equation*}
can be efficiently carried out with a quantum computer (QC). Here $N$ can be exponentially large, $w_k$ are real numbers such that sum $S_w(M)=\sum_{k=0}^{M-1} w_k$ can be calculated in a closed form for any $M$, $S_w(N)=1$ and $f(x)$ is a real function, that is assumed to be easily implementable on a QC. As an application of the technique, we show that Riemann zeta (RZ) function, $\zeta(\sigma+ i t)$ in the critical strip, $\{0 \le \sigma <1, t \in \mathbb{R} \}$, can be obtained in polyLog(t) time. In another setting, we show that RZ function can be obtained with a scaling $t^{1/D}$, where $D \ge 2$ is any integer. These methods provide a vast improvement over the best known classical algorithms; best of which is known to scale as $t^{4/13}$. We present alternative methods to find $\lvert S(f,N) \rvert$ on a QC directly. This method relies on finding the magnitude $A=\lvert \sum_0^{N-1} a_k \rvert$ of a $n$-qubit quantum state with $a_k$ as amplitudes in the computational basis. We present two different ways to do obtain $A$. Finally, a brief discussion of phase/amplitude estimation methods is presented."
[added 12th March 2020]
Y. Nellambakam and K.V.S. Shiv Chaitanya, "Metamaterials and Cesàro convergence" (preprint 01/2020)
[abstract:] "In this paper, we show that the linear dielectrics and magnetic materials in matter obey a special kind of mathematical property known as Ces\`aro convergence. Then, we also show that the analytical continuation of the linear permittivity and permeability to a complex plane in terms of Riemann zeta function. The metamaterials are fabricated materials with a negative refractive index. These materials, in turn, depend on permittivity and permeability of the linear dielectrics and magnetic materials. Therefore, the Ces\`aro convergence property of the linear dielectrics and magnetic materials may be used to fabricate the metamaterials."
[added 12th March 2020]
S. Gorsky, W.A. Britton, R. Zhang, F. Riboli and L. Dal Negro, "Observation of multifractality of light" (preprint 01/2020)
[abstract:] "Many natural patterns and shapes, such as meandering coastlines, clouds, or turbulent flows, exhibit a characteristic complexity mathematically described by fractal geometry. In recent years, the engineering of self-similar structures in photonics and nano-optics technology enabled the manipulation of light states beyond periodic or disordered systems, adding novel functionalities to complex optical media with applications to nano-devices and metamaterials. Here, we extend the reach of fractal 'photonics' by experimentally demonstrating multifractality of light in engineered arrays of dielectric nanoparticles. Our findings stimulate fundamental questions on the nature of transport and localization of wave excitations with multi-scale fluctuations beyond what is possible in traditional fractal systems. Moreover, our approach establishes structure-property relationships that can readily be transferred to planar semiconductor electronics and to artificial atomic lattices, **enabling the exploration of novel quantum phases and many-body effects that emerge directly from fundamental structures of algebraic number theory.**"
[added 12th March 2020]
I. Gálvez-Carrillo, R.M. Kaufmann and A. Tonks, "Three Hopf algebras from number theory, physics & topology, and their common background II: General categorical formulation" (preprint 01/2020)
[abstract:] "We consider three a priori totally different setups for Hopf algebras from number theory, mathematical physics and algebraic topology. These are the Hopf algebra of Goncharov for multiple zeta values, that of Connes–Kreimer for renormalization, and a Hopf algebra constructed by Baues to study double loop spaces. We show that these examples can be successively unified by considering simplicial objects, co-operads with multiplication and Feynman categories at the ultimate level. These considerations open the door to new constructions and reinterpretations of known constructions in a large common framework which is presented step-by-step with examples throughout. In this second part of two papers, we give the general categorical formulation."
[added 12th March 2020]
Y.V. Fyodorov and P. Le Doussal, "Statistics of extremes in eigenvalue-counting staircases" (preprint 01/2020)
[abstract:] "We consider the number $\mathcal{N}(\theta)$ of eigenvalues $e^{i \theta_j}$ of a random unitary matrix, drawn from CUE$_{\beta}(N)$, in the interval $\theta_j \in [\theta_A,\theta]$. The deviations from its mean, $\mathcal{N}(\theta) - \mathbb{E}(\mathcal{N}(\theta))$, form a random process as function of $\theta$. We study the maximum of this process, by exploiting the mapping onto the statistical mechanics of log-correlated random landscapes. By using an extended Fisher--Hartwig conjecture, supplemented with the freezing duality conjecture for log-correlated fields, we obtain the cumulants of the distribution of that maximum for any $\beta >0$. It exhibits combined features of standard counting statistics of fermions (free for $\beta = 2$ and with Sutherland-type interaction for $\beta \ne 2$) in an interval and extremal statistics of the fractional Brownian motion with Hurst index $H=0$. The $\beta = 2$ results are expected to apply to the statistics of zeroes of the Riemann zeta function."
[added 12th March 2020]
W. Yang, "Deligne's conjecture and mirror symmetry" (preprint 01/2020)
[abstract:] "In this paper, we will study the connections between the mirror symmetry of Calabi–Yau threefolds and Deligne's conjecture on the special values of the $L$-functions of critical motives. Using the theory of mirror symmetry, we will develop an explicit method to compute the Deligne's period for a smooth fiber in the mirror family of a one-parameter mirror pair of Calabi–Yau threefolds. We will give two classes of examples to show how this method works and express Deligne's period in terms of the classical periods of the threeform of Calabi–Yau threefolds. In the paper by Candelas, de la Ossa, Elmi and van Straten, they are able to compute the special values of the $L$-function of a Calabi–Yau threefold that is called a rank-$2$ attractor. In this paper, we will compute the Deligne's period of this Calabi–Yau threefold and explicitly show that it satisfies Deligne's conjecture. A second purpose of this paper is to introduce the Deligne's conjecture to the physics community, and provide further evidence that a physics theory can have applications in number theory."
[added 12th March 2020]
M. Nardelli and A. Narelli, "Analyzing some Ramanujan formulas: Mathematical connections with various sectors of black hole physics" (preprint 02/2020)
[abstract:] "The purpose of this paper is to show how using certain mathematical values and/or constants from various Ramanujan expressions, we obtain some mathematical connections with several sectors of black hole physics"
[added 12th March 2020]
CIMPA-CINVESTAV School "p-Adic Numbers, Ultrametric Analysis, and Applications", September 16–24, 2020, Mexico City, Mexico
[added 12th March 2020]
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 RamanujanÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢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 $C(K) = S^3ÃƒÂ¢Ã‹â€ Ã¢â‚¬â€œ(K\times D^2)$ 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 $k$ÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ (SUEBk), a set of incomplete orthonormal special entangled states of ``type $k$ÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ whose complementary space has no special entangled state of ``type $k$ÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢ÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢. 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)ÃƒÂÃ¢â‚¬Â¢= 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 $ÃƒÆ’Ã‚Â¢Ãƒâ€¹Ã¢â‚¬ ÃƒÂ¢Ã¢â€šÂ¬Ã¢â€žÂ¢1$ 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]
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