Freeman Dyson's suggested approach to proving the Riemann Hypothesis using quasi-crystals (from his 2009 AMS lecture)

V. K. Varma, S. Pilati and V. E. Kravtsov, "Conduction in quasi-periodic and quasi-random lattices: Fibonacci, Riemann, and Anderson models" (preprint 07/2016)

[abstract:] "We study the ground state conduction properties of noninteracting electrons in aperiodic but non-random one-dimensional models with chiral symmetry, and make comparisons against Anderson models with non-deterministic disorder. The first model we consider is the Fibonacci lattice, which is a paradigmatic model of quasicrystals; the second is the Riemann lattice, which we define inspired by Dyson's proposal on the possible connection between the Riemann hypothesis and a suitably defined quasicrystal. Our analysis is based on Kohn's many-particle localization tensor defined within the modern theory of the insulating state. In the Fibonacci quasicrystal, where all single-particle eigenstates are critical (i.e., intermediate between ergodic and localized), the noninteracting electron gas is found to be a conductor at most electron densities, including the half-filled case; however, at various specific fillings $\rho$, including the values $\rho = 1/\rho^n$, where $g$ is the golden ratio and $n$ is any integer, the gas turns into an insulator due to spectral gaps. Metallic behaviour is found at half-filling in the Riemann lattice as well; however, in contrast to the Fibonacci quasicrystal, the Riemann lattice is generically an insulator due to single-particle eigenstate localization, likely at all other fillings. Its behaviour turns out to be alike that of the off-diagonal Anderson model, albeit with different system-size scaling of the band-centre anomalies. The advantages of analysing the Kohn's localization tensor instead of other measures of localization familiar from the theory of Anderson insulators (such as the participation ratio or the Lyapunov exponent) are highlighted."

K. D. Thomas, "From prime numbers to nuclear physics and beyond", The Institute Letter (spring 2013) pp.2, 8–9

P.A.M. Dirac, "The relation between mathematics and physics" (lecture delivered on presentation of the James Scott prize, 6 February 1939)

[excerpt:] "There is thus a possibility that the ancient dream of philosophers to connect all Nature with the properties of whole numbers will some day be realized. To do so physics will have to develop a long way to establish the details of how the correspondence is to be made. One hint for this development seems pretty obvious, namely, the study of whole numbers in modern mathematics is inextricably bound up with the theory of functions of a complex variable, which theory we have already seen has a good chance of forming the basis of the physics of the future. The working out of this idea would lead to a connection between atomic theory and cosmology."

D. Schumayer and D.A.W. Hutchinson, "Physics of the Riemann hypothesis", Rev. Mod. Phys. 83 (2011) 307–330

[abstract:] "Physicists become acquainted with special functions early in their studies. Consider our perennial model, the harmonic oscillator, for which we need Hermite functions, or the Laguerre functions in quantum mechanics. Here we choose a particular number theoretical function, the Riemann zeta function and examine its influence in the realm of physics and also how physics may be suggestive for the resolution of one of mathematics' most famous unconfirmed conjectures, the Riemann Hypothesis. Does physics hold an essential key to the solution for this more than hundred-year-old problem? In this work we examine numerous models from different branches of physics, from classical mechanics to statistical physics, where this function plays an integral role. We also see how this function is related to quantum chaos and how its pole-structure encodes when particles can undergo Bose–Einstein condensation at low temperature. Throughout these examinations we highlight how physics can perhaps shed light on the Riemann Hypothesis. Naturally, our aim could not be to be comprehensive, rather we focus on the major models and aim to give an informed starting point for the interested Reader."

M. Wolf, "Will a physicists prove the Riemann Hypothesis?" (preprint 10/2014)

[abstract:] "In the first part we present the number theoretical properties of the Riemann zeta function and formulate the Riemann Hypothesis. In the second part we review some physical problems related to this hypothesis: the links with random matrix theory, relation with the Lee–Yang theorem on the zeros of the partition function, random walks, billiards etc."

L. Ionescu, "Remarks on physics as number theory" (preprint 08/2011)

[abstract:] "There are numerous indications that physics, at its foundations, is algebraic number theory.

The Bohr model for the hydrogen atom is the starting point of a quantum computing model on serial-parallel graphs is provided as the quantum system affording the partition function of the Riemann gas/primon model. The propagator of the corresponding discrete path integral formalism is a fermionic zeta value 'closely' related to the experimental value of the fine structure constant corresponding to the continuum path integral formalism of Feynman.

The duality of multiplicative number theory, as a theory of the graded Hopf module of integers, and the Kleinian geometry of the primary finite fields underlying its base of primitive elements, are briefly mentioned in this framework ('Integer CFT')."

A. LeClair, "An electrostatic depiction of the validity of the Riemann Hypothesis" (preprint 05/2013)

[abstract:] "We construct an electric field from the real and imaginary parts of an entire function $\xi(z)$ which arises in the quantum statistical mechanics of relativistic gases when the spatial dimension $d$ is analytically continued into the complex $z$ plane. This function is built from the Gamma and Riemann zeta functions and is known to satisfy the functional identity $\xi(z) = \xi(1-z)$. We describe how this non-trivial identity can be demonstrated using quantum field theory arguments in a cylindrical geometry, where it relates finite temperature black body physics to the Casimir energy on a circle. The structure of the electric field in the critical strip is determined by its behavior near the Riemann zeros on the critical line $\Re(z) = 1/2$, where each zero can be assigned a plus or minus vorticity of a related pseudo-magnetic field. Using these properties, we show that a hypothetical Riemann zero that is off the critical line leads to a frustration of the electric field, which is to say, an incompatibility with the electric field pattern that is a consequence of the infinite number of zeros along the critical line."

G. França and A. LeClair, "On the validity of the Euler product inside the critical strip" (preprint 10/2014)

[abstract:] "The Euler Product Formula relates Riemann's zeta function $\zeta(s)$ to an infinite product over primes, and is known to be valid for $\Re(s) > 1$. We provide arguments that the formula is actually valid for $\Re(s) > 1/2$ and $\Im(s)\neq 0$ due to the conditional convergence of the infinite product in this regime. The argument relies on four ingredients: the prime number theorem, an Abel transform, a central limit theorem for the Random Walk of the Primes series $\sum_p\cos(t\log p)$, where $p$ is a prime number, and the Cauchy criterion for convergence. The significance of $\Re(s) > 1/2$ arises from the universality of the $N^{1/2}$ growth of fluctuations in various central limit theorems for independent and weakly dependent random processes, which are common in statistical physics for systems of size $N$. Numerical evidence of this surprising result is presented, and some of its consequences are discussed."

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 con- centrations. The mixed chemical potential is converted to the time domain of an action poten- tial 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."

V. Blomer, J. Bourgain and Z. Rudnick, "Small gaps in the spectrum of the rectangular billiard" (preprint 04/2016)

"We study the size of the minimal gap between the first $N$ eigenvalues of the Laplacian on a rectangular billiard having irrational squared aspect ratio $\alpha$, in comparison to the corresponding quantity for a Poissonian sequence. If $\alpha$ is a quadratic irrationality of certain type, such as the square root of a rational number, we show that the minimal gap is roughly of size $1/N$, which is essentially consistent with Poisson statistics. We also give related results for a set of $\alpha$'s of full measure. However, on a fine scale we show that Poisson statistics is violated for all $\alpha$. The proofs use a variety of ideas of an arithmetical nature, involving Diophantine approximation, the theory of continued fractions, and results in analytic number theory. One of our results is conditional on the Riemann Hypothesis."

J. Mayoh and A.M. García-García, "Number theory, periodic orbits and superconductivity in nano-cubes" (preprint 04/2014)

[abstract:] "We combine mean field, number theory and semi-classical techniques to study superconductivity in isolated superconducting nano-cubes and nano-squares of size $L$ in the limit. We study the case in which disorder is negligible and $k_FL\gg 1 where$k_F$is the Fermi wave vector. By using periodic orbit theory we find an explicit analytical expression of the size dependence of the superconducting order parameter that takes into account contributions from both the spectral density and the matrix elements related to density fluctuations of the one-body eigenstates. We employ number theory techniques to compute analytically the average size dependence of the matrix elements. The leading size dependence of the matrix elements seems to be universal as it agrees with that found in chaotic grains. For sizes$L \gtrsim 10$nm there is very good agreement between numerical and analytical results. For the parameters that describe conventional metallic superconductors deviations from the bulk limit are still important for grains as large as$L\sim 50$nm." J. Peng, S. Sun, V.K. Narayana, V.J. Sorger and T. El-Ghazawi, Integrated nanophotonics architecture for residue number system arithmetic" (preprint 11/2017) [abstract:] "Residue number system (RNS) enables dimensionality reduction of an arithmetic problem by representing a large number as a set of smaller integers, where the number is decomposed by prime number factorization using the moduli as basic functions. These reduced problem sets can then be processed independently and in parallel, thus improving computational efficiency and speed. Here we show an optical RNS hardware representation based on integrated nanophotonics. The digit-wise shifting in RNS arithmetic is expressed as spatial routing of an optical signal in$2\times 2$hybrid photonic-plasmonic switches. Here the residue is represented by spatially shifting the input waveguides relative to the routers outputs, where the moduli are represented by the number of waveguides. By cascading the photonic$2\times 2$switches, we design a photonic RNS adder and a multiplier forming an all-to-all sparse directional network. The advantage of this photonic arithmetic processor is the short (10's ps) computational execution time given by the optical propagation delay through the integrated nanophotonic router. Furthermore, we show how photonic processing in-the-network leverages the natural parallelism of optics such as wavelength-division-multiplexing or optical angular momentum in this RNS processor. A key application for photonic RNS is the functional analysis convolution with widespread usage in numerical linear algebra, computer vision, language- image- and signal processing, and neural networks." G. França and A. LeClair, "Statistical and other properties of Riemann zeros based on an explicit formula for the$n$-th zero" (preprint 07/2013) [abstract:] "Very recently, a transcendental equation satisfied by the$n$-th Riemann zero on the critical line was derived by one of us. Here we provide a more detailed analysis of this result, demonstrating more rigorously that the Riemann zeros occur on the critical line with real part equal to 1/2, and their imaginary parts satisfy such a transcendental equation. From this equation, the counting of zeros on the critical line can be derived, yielding precisely the Riemann–von Mangoldt counting function$N(T)$for the zeros on the entire critical strip. Therefore, these results constitute a proposal for establishing the validity of the Riemann Hypothesis. We also obtain an approximate solution of the transcendental equation, in closed form, based on the Lambert$W$function. This yields a very good estimate for the Riemann zeros, for arbitrarily high values on the critical line. We then obtain numerical solutions of the complete transcendental equation, yielding accurate numbers for the Riemann zeros, which agree with previous known results found in the literature. Employing these numerical solutions, we verify that they are accurate enough to confirm Montgomery's pair correlation conjecture and also to reconstruct the prime number counting formula." C. Forestiere, G.F. Walsh, G. Miano and L. Dal Negro, "Nanoplasmonics of prime number arrays", Optics Express 17 (26) (2009) 24288–24303 [abstract:] "In this paper, we investigate the plasmonic near-field localization and the far-field scattering properties of non-periodic arrays of Ag nanoparticles generated by prime number sequences in two spatial dimensions. In particular, we demonstrate that the engineering of plasmonic arrays with large spectral flatness and particle density is necessary to achieve a high density of electromagnetic hot spots over a broader frequency range and a larger area compared to strongly coupled periodic and quasi-periodic structures. Finally, we study the far-field scattering properties of prime number arrays illuminated by plane waves and we discuss their angular scattering properties. The study of prime number arrays of metal nanoparticles provides a novel strategy to achieve broadband enhancement and localization of plasmonic fields for the engineering of nanoscale nano-antenna arrays and active plasmonic structures." G. Seiden, "Number-theoretic expressions obtained through analogy between prime factorization and optical interferometry", Physical Review A 85 (2012) 043842 [abstract:] "Prime factorization is an outstanding problem in arithmetic with important consequences in a variety of fields, most notably cryptography. Here I employ the intriguing analogy between prime factorization and optical interferometry in order to obtain analytic expressions for closely related functions, including the number of distinct prime factors." P. Rosakis, "Continuum Surface Energy from a Lattice Model" (preprint 01/2012) [abstract:] "The energy of a homogeneously deformed, faceted crystal is calculated in the context of a central force lattice model in two dimensions. It is shown that the energy equals the bulk elastic energy, plus the integral over the boundary of a surface energy density, plus the sum over the vertices of a corner energy function. This is an exact result when the interatomic potential has finite range; for an infinite-range potential it is asymptotically valid as the lattice parameter tends to zero. The surface energy density is obtained explicitly as a function of the deformation gradient and boundary normal. The corner energy is found as an explicit function of the deformation gradient and the normals of the two facets meeting at the corner. A new bond counting approach is used, which allows the problem to be reduced to the well known lattice point problem of number theory." M. Nixon, M. Fridman, E. Ronen, A.A. Friesem, N. Davidson and I. Kanter, "Controlling synchronization in large laser networks using number theory" (preprint 12/2011) [abstract:] "Synchronization in networks with delayed coupling are ubiquitous in nature and play a key role in almost all fields of science including physics, biology, ecology, climatology and sociology. In general, the published works on network synchronization are based on data analysis and simulations, with little experimental verification. Here we develop and experimentally demonstrate various multi-cluster phase synchronization scenarios within coupled laser networks. Synchronization is controlled by the network connectivity in accordance to number theory, whereby the number of synchronized clusters equals the greatest common divisor of network loops. This dependence enables remote switching mechanisms to control the optical phase coherence among distant lasers by local network connectivity adjustments. Our results serve as a benchmark for a broad range of coupled oscillators in science and technology, and offer feasible routes to achieve multi-user secure protocols in communication networks and parallel distribution of versatile complex combinatorial tasks in optical computers." I.D. Mayergoyz, "Plasmon Resonances in Nanoparticles, Their Applications to Magnetics and Relation to the Riemann Hypothesis" (preprint 04/2011) [abstract:] "The review of the mathematical treatment of plasmon resonances as an eigenvalue problem for specific boundary integral equations is presented and general properties of plasmon spectrum are outlined. Promising applications of plasmon resonances to magnetics are described. Interesting relation of eigenvalue treatment of plasmon resonances to the Riemann hypothesis is established." S. Boatto and J. Koiller, "Vortices on closed surfaces" (preprint 02/2008) [abstract:] "We consider$N$point vortices$s_j$of strengths$\kappa_j$moving on a closed (compact, boundaryless, orientable) surface$S$with riemannian metric$g$. As far as we know, only the sphere or surfaces of revolution, the latter qualitatively, have been treated in the available literature. The aim of this note is to present an intrinsic geometric formulation for the general case. We give a simple proof of Kimura's conjecture that a dipole describes geodesic motion. Searching for integrable vortex pairs systems on Liouville surfaces is in order. The vortex pair system on a triaxial ellipsoid extends Jacobi's geodesics. Is it Arnold–Liouville integrable? Not in our wildest dreams is another possibility: that quantizing a vortex system could relate with a million dollars worth question, but we took courage – nerve is more like it – to also present it." [The paper concludes with the following suggestion that there might be a connection to the Riemann Hypothesis: "A connection may exist between the zeros of Riemann's zeta function and the quantization of a 3/2 degrees of freedom vortex monopole problem on some compact Riemann surface..."] Y. Hashimoto, "Correlations of multiplicities in length spectra for congruence subgroups" (preprint 02/2012) [abstract:] "Bogomolny-Leyvraz-Schmit (1996) and Peter (2002) proposed an asymptotic formula for the correlation of the multiplicities in length spectrum on the modular surface, and Lukianov (2007) extended its asymptotic formula to the Riemann surfaces derived from the congruence subgroup$\Gam_0(n)$and the quaternion type co-compact arithmetic groups. The coefficients of the leading terms in these asymptotic formulas are described in terms of Euler products over prime numbers, and they appear in eigenvalue statistic formulas found by Rudnick (2005) and Lukianov (2007) for the Laplace-Beltrami operators on the corresponding Riemann surfaces. In the present paper, we further extend their asymptotic formulas to the higher level correlations of the multiplicities for any congruence subgroup of the modular group." R. Movassagh, Y. Tsuji and R. Hoffmann, "The exact form of the Green's function of the Hückel (tight binding) model" (preprint 07/2014) [abstract:] "The applications of the Hückel (tight binding) model are ubiquitous in quantum chemistry and solid state physics. The matrix representation is isomorphic to an unoriented vertex adjacency matrix of a bipartite graph, which is also the Laplacian matrix plus twice the identity. In this paper, we analytically calculate the determinant and, when it exists, the inverse of this matrix in connection with the Green's function,$G$, of the$N \times N$Hückel matrix for linear chains and cyclic systems. For an open linear chain we prove that$G$is a real symmetric matrix whose entries are$G(r,s) = (-1)^{\frac{r+s-1}{2}}$when$r$is even and$s < r$is odd;$G(r,s)=0$otherwise. A corollary is a closed form expression for a Harmonic sum. For a ring we calculate the inverse, give formulas for the entries and find that it is always a Toeplitz matrix. We then extend the results to$d$-dimensional lattices, whose linear size is$N$. The existence of the inverse becomes a question of number theory. We prove that the inverse exists if and only if$N+1$and$d$are odd and$d$is smaller than the smallest divisor of$N+1$. We corroborate our results by numerical demonstrations of the entry patterns of the Green's function and discuss applications related to transport and conductivity." "A Trillion Triangles", AIM News (22/09/09) "In addition to the practical advances required for this result, the answer also has theoretical implications. According to mathematician Michael Rubinstein from the University of Waterloo, "A few years ago we combined ideas from number theory and physics to predict how congruent numbers behave statistically. I was very pleased to see that our prediction was quite accurate."" K. Nakayama, F. Takahashi and T.T. Yanagida, "Number theory dark matter" (preprint 02/2011) [abstract:] "We propose that the stability of dark matter is ensured by a discrete subgroup of the U(1)B-L gauge symmetry, Z_2(B-L). We introduce a set of chiral fermions charged under the U(1)B-L in addition to the right-handed neutrinos, and require the anomaly-cancellation conditions associated with the U(1)B-L gauge symmetry. We find that the possible number of fermions and their charges are tightly constrained, and that non-trivial solutions appear when at least five additional chiral fermions are introduced. The Fermat theorem in the number theory plays an important role in this argument. Focusing on one of the solutions, we show that there is indeed a good candidate for dark matter, whose stability is guaranteed by Z_2(B-L)." Bo Hu, "Neutrino mixing and discrete symmetries" (preprint 12/2012) [abstract:] "A model independent study of neutrino mixing based on a new method to derive mixing patterns is presented. An interesting result we find is that, in the case where unbroken residual symmetries of the Majorana neutrino and left-handed charged-lepton mass matrices obey some general assumptions, the complete set of possible mixing patterns can be determined by the solutions to the constraint equation with the help of algebraic number theory. This method can also be applied to more general cases beyond the minimal scenario. Several applications and phenomenological implications are discussed." C. King, "Experimental observations on the Riemann hypothesis, and the Collatz conjecture" (preprint 05/2010) [abstract:] "This paper seeks to explore whether the Riemann hypothesis falls into a class of putatively unprovable mathematical conjectures, which arise as a result of unpredictable irregularity. It also seeks to provide an experimental basis to discover some of the mathematical enigmas surrounding these conjectures, by providing Matlab and C programs which the reader can use to explore and better understand these systems." C. Moore and A. Russell, "Approximate representations and approximate homomorphisms" (preprint 09/2010) [abstract:] "Approximate algebraic structures play a defining role in arithmetic combinatorics and have found remarkable applications to basic questions in number theory and pseudorandomness. Here we study approximate representations of finite groups: functions f:G -> U_d such that Pr[f(xy) = f(x) f(y)] is large, or more generally Exp_{x,y} ||f(xy) - f(x)f(y)||^2 is small, where x and y are uniformly random elements of the group G and U_d denotes the unitary group of degree d. We bound these quantities in terms of the ratio d / d_min where d_min is the dimension of the smallest nontrivial representation of G. As an application, we bound the extent to which a function f : G -> H can be an approximate homomorphism where H is another finite group. We show that if H's representations are significantly smaller than G's, no such f can be much more homomorphic than a random function. We interpret these results as showing that if G is quasirandom, that is, if d_min is large, then G cannot be embedded in a small number of dimensions, or in a less-quasirandom group, without significant distortion of G's multiplicative structure. We also prove that our bounds are tight by showing that minors of genuine representations and their polar decompositions are essentially optimal approximate representations." B. Luque, O. Miramontes and L. Lacasa, "Number theoretic example of scale-free topology inducing self-organized criticality", Phys. Rev. Lett. 101 (2008) 158702 [abstract:] "In this Letter we present a general mechanism by which simple dynamics running on networks become self-organized critical for scale-free topologies. We illustrate this mechanism with a simple arithmetic model of division between integers, the division model. This is the simplest self-organized critical model advanced so far, and in this sense it may help to elucidate the mechanism of self-organization to criticality. Its simplicity allows analytical tractability, characterizing several scaling relations. Furthermore, its mathematical nature brings about interesting connections between statistical physics and number theoretical concepts. We show how this model can be understood as a self-organized stochastic process embedded on a network, where the onset of criticality is induced by the topology." D. Vilone, "Complex network approach to number theory" (preprint 09/2014) [abstract:] "In this short paper, following the most recent advances in complex network theory, a new approach to number theory with potential applications to other fields is proposed. The model by Garcia-Perez, Serrano and Boguna, introduces an algorithm which allows to create a bipartite graph of integers (with primes and composites) statistically very close to the real one. Since the algorithm is defined a priori, we can have a description of the simulated prime number distribution in terms of a known differential equation, which in general can be treated more easily. The so determined properties of the simulated distribution can give useful hints about the behavior of the real prime number distribution. In principle it could be also possible to demonstrate open questions in number theory, proven the total equivalence of the simulated and real distributions. I show that the model by Garcia-Perez, Serrano and Boguna, though very good to describe the architecture of the relations among composites and primes, can not be useful to catch the most subtle properties of prime numbers. Anyway, the path suggested by such model is very promising, and should drive researchers to look for more refined algorithms to simulate the structure of integer numbers." V.L.Cartas, "The Riemann zeta function applied to the glassy systems and neural networks" (presented at International Conference on Theoretical Physics, Paris, 22–27 July 2002) [Abstract:] "In the present paper it is described how the Riemann zeta function could be a very useful tool in the analyze of the glassy systems and the neural networks. According to A. Crisanti and F. Ritort, this kind of complex systems could be analyzed using a simple solvable model of glass: "The oscillator model" which is defined by a set of N non-interacting harmonic oscillators with energy. The Riemann zeta function is used to describe the Crisanti-Ritort System. It has been also made a topological study in order to have a more intuitive representation of the critical points, where the states of the system changes." P. Le Doussal, Z. Ristivojevic and K.J. Wiese, "Exact form of the exponential correlation function in the glassy super-rough phase" (preprint 04/2013) [abstract:] "We consider the random-phase sine-Gordon model in two dimensions. It describes two-dimensional elastic systems with random periodic disorder, such as pinned flux-line arrays, random field XY models, and surfaces of disordered crystals. The model exhibits a super-rough glass phase at low temperature$T < T_{c}$with relative displacements growing with distance$r$as$\bar{\langle [\theta(r)-\theta(0)]^2\rangle} \simeq A(\tau) \ln^2 (r/a)$, where$A(\tau) = 2 \tau^2- 2 \tau^3 +\mathcal{O}(\tau^4)$near the transition and$\tau=1-T/T_{c}$. We calculate all higher cumulants and show that they grow as$\bar{\langle[\theta(r)-\theta(0)]^{2n}\rangle}_c \simeq [2 (-1)^{n+1} (2n)! \zeta(2n-1) \tau^2 + \mathcal{O}(\tau^3) ] \ln(r/a)$,$n \geq 2$, where$\zeta$is the Riemann zeta function. By summation we obtain the decay of the exponential correlation function as$\bar{\langle e^{iq\left[\theta(r)-\theta(0)\right]}\rangle} \simeq (a/r)^{\eta(q)} \exp(-\frac{1}{2}\mathcal{A}(q)\ln^2 (r/a))$where$\eta(q)$and${\cal A}(q)$are obtained for arbitrary$q \leq 1$to leading order in$\tau$. The anomalous exponent is$\eta(q) = c q^2 - \tau^2 q^2 [2\gamma_E+\psi(q)+\psi(-q)]$in terms of the digamma function$\psi$, where$c$is non-universal and$\gamma_E$is the Euler constant. The correlation function shows a resonant deep at$q=1$, corresponding to fermion operators in the dual picture, which should bevisible in Bragg scattering experiments." D. Merlini, "The Riemann magneton of the primes" (preprint 04/04) [abstract:] "We present a calculation involving a function related to the Riemann Zeta function and suggested by two recent works concerning the Riemann Hypothesis: one by Balazard, Saias and Yor and the other by Volchkov. We define an integral m(r) involving the Zeta function in the complex variable s = r + it and find a particularly interesting expression for m(r) which is rigorous at least in some range of r. In such a range we find that there are two discontinuities of the derivative m'(r) at r = 1 and r = 0, which we calculate exactly. The jump at r = 1 is given by 4*Pi. The validity of the expression for m(r) up to r = 1/2 is equivalent to the truth of the Riemann Hypothesis (RH). Assuming RH the expression for m (r) gives m = 0 at r = 1/2 and the slope m'(r) = Pi*(1 + gamma) = 4.95 at r = 1/2 (where gamma = 0.577215... is the Euler constant). As a consequence, if the expression for m(r) can be continued up to r = 1/2, then if we interpret m(r) as a magnetization in the presence of a magnetic field h = r - 1/2 (or as a "free energy" at inverse temperature beta proportional to r), there is a first order phase transition at r = 1/2 (h = 0) with a jump of m'(r) given by 2*Pi times the first Lin coefficient lambda_1 = [1+gamma/2-(1/2)ln(4*Pi)] = 0.0230957. Independently of the RH, by looking at the behavior of the convergent Taylor expansion of m(r) at r = 1-, m(r = 1/2+) as well as the first Lin coefficient may be evaluated using the Euler product formula, in terms of the primes. This gives further evidence for the possible truth of the Riemann Hypothesis." S. Beltraminelli, D. Merlini, S. Sekatskii, "A hidden symmetry related to the Riemann Hypothesis with the primes into the critical Strip" (preprint 03/2008) [abstract:] "In this note concerning integrals involving the logarithm of the Riemann Zeta function, we extend some treatments given in previous pioneering works on the subject and introduce a more general set of Lorentz measures. We first obtain two new equivalent formulations of the Riemann Hypothesis (RH). Then with a special choice of the measure we formulate the RH as a 'hidden symmetry', a global symmetry which connects the region outside the critical strip with that inside the critical strip. The Zeta function with all the primes appears as argument of the Zeta function in the critical strip. We then illustrate the treatment by a simple numerical experiment. The representation we obtain go a little more in the direction to believe that RH may eventually be true." H. Kösters, "The Riemann zeta-function and the sine kernel" (preprint 03/2008) [abstract:] "We point out an interesting occurrence of the sine kernel in connection with the shifted moments of the Riemann zeta-function along the critical line. We establish this occurrence rigorously for the shifted second moment and, under some constraints on the shifts, for the shifted fourth moment. Our proofs of these results closely follow the classical proofs for the non-shifted moments of the Riemann zeta-function. Furthermore, we conjecture that the sine kernel also occurs in connection with the higher (even) shifted moments and show that this conjecture is closely related to a recent conjecture by Conrey, Farmer, Keating, Rubinstein, and Snaith." G.A. Hiary and M.O. Rubinstein, "Uniform asymptotics for the full moment conjecture of the Riemann zeta function" (preprint 06/2011) [abstract:] "Conrey, Farmer, Keating, Rubinstein, and Snaith recently conjectured formulas for the full asymptotics of the moments of$L$-functions. In the case of the Riemann zeta function, their conjecture states that the$2k$-th absolute moment of zeta on the critical line is asymptotically given by a certain$2k$-fold residue integral. This residue integral can be expressed as a polynomial of degree$k^2$, whose coefficients are given in exact form by elaborate and complicated formulas. In this article, uniform asymptotics for roughly the first$k$coefficients of the moment polynomial are derived. Numerical data to support our asymptotic formula are presented. An application to bounding the maximal size of the zeta function is considered." A. Pichler, "On a rapidly converging series for the Riemann zeta function" (preprint 01/2012) [abstract:] "To evaluate Riemann's zeta function is important for many investigations related to the area of number theory, and to have quickly converging series at hand in particular. We investigate a class of summation formulae and find, as a special case, a new proof of a rapidly converging series for the Riemann zeta function. The series converges in the entire complex plane, its rate of convergence being significantly faster than comparable representations, and so is a useful basis for evaluation algorithms. The evaluation of corresponding coefficients is not problematic, and precise convergence rates are elaborated in detail. The globally converging series obtained allow to reduce Riemann's hypothesis to similar properties on polynomials. And interestingly, Laguerre's polynomials form a kind of leitmotif through all sections." R. Fisch, "Evidence of long range order in the Riemann zeta function" (preprint, 10/2012) [abstract:] "We have done a statistical analysis of some properties of the contour lines$\Im(\zeta(s)) = 04 of the Riemann zeta function. We find that this function is broken up into strips whose average width on the critical line does not appear to vary with height. We also compute the position of the primary zero for the lowest 200 strips, and find that this probability distribution also appears to be scale invariant."

N. Barros e Sa and I. Bengtsson, "Families of complex Hadamard matrices" (preprint 02/2012)

[abstract:] "What is the dimension of a smooth family of complex Hadamard matrices including the Fourier matrix? We address this problem with a power series expansion. Studying all dimensions up to 100 we find that the first order result is misleading unless the dimension is 6, or a power of a prime. In general the answer depends critically on the prime number decomposition of the dimension. Our results suggest that a general theory is possible. We discuss the case of dimension 12 in detail, and argue that the solution consists of two 13-dimensional families intersecting in a previously known 9-dimensional family. A precise conjecture for all dimensions equal to a prime times another prime squared is formulated."

R.C. McPhedran, "Constructing a proof of the Riemann Hypothesis" (preprint 08/2013)

[abstract:] "This paper compares the distribution of zeros of the Riemann zeta function $\zeta(s)$ with those of a symmetric combination of zeta functions, denoted ${\cal T}_+(s)$, known to have all its zeros located on the critical line $\Re(s)=1/2$. Criteria are described for constructing a suitable quotient function of these, with properties advantageous for establishing an accessible proof that $\zeta(s)$ must also have all its zeros on the critical line: the celebrated Riemann hypothesis. While the argument put forward is not at the level of rigour required to constitute a full proof of the Riemann hypothesis, it should convince non-specialists that it must hold."

R.C. McPhedran, "The Riemann Hypothesis for Dirichlet $L$ functions" (preprint 08/2013)

[abstract:] "This paper studies the connections between the zeros and their distribution functions for two particular Dirichlet $L$ functions: the Riemann zeta function, and the Catalan beta function, also known as the Dirichlet beta function. It is shown that the Riemann hypothesis holds for the Dirichlet beta function $L_{-4}(s)$ if and only if it holds for $\zeta (s)$ - a particular case of the Generalized Riemann Hypothesis."

R.C. McPhedran and C.G. Poulton, The Riemann Hypothesis for symmetrised combinations of zeta functions" (preprint 08/2013)

[abstract:] "This paper studies combinations of the Riemann zeta function, based on one defined by P.R. Taylor, which was shown by him to have all its zeros on the critical line. With a rescaled complex argument, this is denoted here by ${\cal T}_-(s)$, and is considered together with a counterpart function ${\cal T}_+(s)$, symmetric rather than antisymmetric about the critical line. We prove that ${\cal T}_+(s)$ has all its zeros on the critical line, and that the zeros of both functions are all of first order. We establish a link between the zeros of ${\cal T}_-(s)$ and of ${\cal T}_+(s)$ with those of the zeros of the Riemann zeta function $\zeta(2 s-1)$, which enables us to prove that, if the Riemann hypothesis holds, then the distribution function of the zeros of $\zeta (2 s-1)$ agrees with those for ${\cal T}_-(s)$ and of ${\cal T}_+(s)$ in all terms which do not remain finite as $t\rightarrow \infty$."

R.C. McPhedran, L.C. Botten, D.J. Williamson, N.-A.P. Nicorovici, "The Riemann hypothesis for angular lattice sums (preprint 07/2010)

[abstract:] "We present further results on a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle, supplementing those in a previous paper (McPhedran {\em et al}, 2008). We give a general expression which permits numerical evaluation of members of the class of sums to arbitrary order. We use this to illustrate numerically the properties of trajectories along which the real and imaginary parts of the sums are zero, and we show results for the first two of a particular set of angular sums (denoted ${\cal C}(1,4 m;s)$) which indicate their density of zeros on the critical line of the complex exponent is the same as that for the product (denoted ${\cal C}(0,1;s)$) of the Riemann zeta function and the Catalan beta function. We then introduce a function which is the quotient of the angular lattice sums ${\cal C}(1,4 m;s)$ with ${\cal C}(0,1;s)$, and use its properties to prove that ${\cal C}(1,4 m;s)$ obeys the Riemann hypothesis for any $m$ if and only if ${\cal C}(0,1;s)$ obeys the Riemann hypothesis. We furthermore prove that if the Riemann hypothesis holds, then ${\cal C}(1,4 m;s)$ and ${\cal C}(0,1;s)$ have the same distribution of zeros on the critical line (in a sense made precise in the proof)."

R.C. McPhedran, L.C. Botten, N.-A.P. Nicorovici, "Further results on the Riemann hypothesis for angular lattice sums" (preprint 07/2009)

[abstract:] "We present further results on a class of sums which involve complex powers of the distance to points in a two-dimensional square lattice and trigonometric functions of their angle, supplementing those in a previous paper (McPhedran et al, Proc. Roy. Soc., 2008). We give a general expression which permits numerical evaluation of members of the class of sums to arbitrary order. We use this to illustrate numerically the properties of trajectories along which the real and imaginary parts of the sums are zero, and we show results for the first two of a particular set of angular sums which indicate their density of zeros on the critical line of the complex exponent is the same as that for the product of the Riemann zeta function and the Catalan beta function."

R. C. McPhedran, "Zeros of lattice sums: 2. A geometry for the Generalised Riemann Hypothesis" (preprint 02/2016)

[abstract:] "The location of zeros of the basic double sum over the square lattice is studied. This sum can be represented in terms of the product of the Riemann zeta function and the Dirichlet beta function, so that the assertion that all its non-trivial zeros lie on the critical line is a particular case of the Generalised Riemann Hypothesis (GRH). It is shown that a new necessary and sufficient condition for this special case of the GRH to hold is that a particular set of equimodular and equiargument contours of a ratio of MacDonald function double sums intersect only on the critical line. It is further shown that these contours could only intersect off the critical line on the boundary of discrete regions of the complex plane called 'inner islands'. Numerical investigations are described related to this geometrical condition, and it is shown that for the first ten thousand zeros of both the zeta function and the beta function over 70% of zeros lie outside the inner islands, and thus would be guaranteed to lie on the critical line by the arguments presented here. A new sufficient condition for the Riemann Hypothesis to hold is also presented."

D.W. Farmer and S.M. Gonek, "Pair correlation of the zeros of the derivative of the Riemann $\xi$-function" (preprint 03/2008)

[abstract:] "The complex zeros of the Riemann zeta-function are identical to the zeros of the Riemann xi-function, $\xi(s)$. Thus, if the Riemann Hypothesis is true for the zeta-function, it is true for $\xi(s)$. Since $\xi(s)$ is entire, the zeros of $\xi'(s)$, its derivative, would then also satisfy a Riemann Hypothesis. We investigate the pair correlation function of the zeros of $\xi'(s)$ under the assumption that the Riemann Hypothesis is true. We then deduce consequences about the size of gaps between these zeros and the proportion of these zeros that are simple."

J.B. Conrey, N.C. Snaith, "Correlations of eigenvalues and Riemann zeros" (preprint 03/2008)

[abstract:] "We present a new approach to obtaining the lower order terms for $n$-correlation of the zeros of the Riemann zeta function. Our approach is based on the `ratios conjecture' of Conrey, Farmer, and Zirnbauer. Assuming the ratios conjecture we prove a formula which explicitly gives all of the lower order terms in any order correlation. Our method works equally well for random matrix theory and gives a new expression, which is structurally the same as that for the zeta function, for the $n$-correlation of eigenvalues of matrices from U(N)."

E. Dueñez, D.W. Farmer, S. Froehlich, C. Hughes, F. Mezzadri and T. Phan, "Roots of the derivative of the Riemann zeta function and of characteristic polynomials" (preprint 02/2010)

[abstract:] "We investigate the horizontal distribution of zeros of the derivative of the Riemann zeta function and compare this to the radial distribution of zeros of the derivative of the characteristic polynomial of a random unitary matrix. Both cases show a surprising bimodal distribution which has yet to be explained. We show by example that the bimodality is a general phenomenon. For the unitary matrix case we prove a conjecture of Mezzadri concerning the leading order behavior, and we show that the same follows from the random matrix conjectures for the zeros of the zeta function."

S. H. Saker, "Large spaces between the zeros of the Riemann Zeta-Function" (preprint 06/09)

[abstract:] "On the hypothesis that the mixed moments of Hardy's function and its derivative are correctly predicted by random matrix theory we derive new large spaces between the zeros of the Riemann zeta-function. Our proof depends on new Wirtinger-type inequalities and numerical solutions of algebraic equations."

M. Fujimoto and K. Uehara, "A Brief note on the Riemann hypothesis" (preprint 06/09)

[abstract:] "We deal with the Euler's alternating series of the Riemann zeta function to define a regularized ratio appeared in the functional equation even in the critical strip and show some evidence to indicate the hypothesis in this note."

M. Fujimoto, K. Uehara, "A brief note on the Riemann hypothesis II" (preprint 03/2010)

[abstract:] "We deal with the Euler's alternating series of the Riemann zeta function to define a finite ratio from divergent quantities appeared in the functional equation even in the critical strip and show some evidence to indicate the hypothesis in this note."

G.K. Óttarsson, "Discrete calculus and elliptic functions" (preprint 01/2014)

[abstract:] "In a paper from 18 August 2001 available at www.islandia.is/gko/010818.pdf, a thermoelectric generator was constructed from a large number of series connected parallelepipeds. The hot and/or cold reservoir was made of some electrically conductive metal, and the fluid was to some extent conductive to the electrical ground. This topology generated a number of small capacitors, each formed by two parallelepiped crystal faces and the grounded thermal reservoir. When analysing the frequency behaviour of such a device, rational polynomials manifested themselves and proved to be a rich source of advanced mathematical relations."

These relations involve the Riemann zeta function, Bernoulli numbers, the Gamma function, Euler's constant and Stirling's Formula.

G. Chinta, J. Jorgenson and A. Karlsson, "Complexity and heights of tori" (preprint 10/2011)

[abstract:] "We prove detailed asymptotics for the number of spanning trees, called complexity, for a general class of discrete tori as the parameters tend to infinity. The proof uses in particular certain ideas and techniques from an earlier paper. Our asymptotic formula provides a link between the complexity of these graphs and the height of associated real tori, and allows us to deduce some corollaries on the complexity thanks to certain results from analytic number theory. In this way we obtain a conjectural relationship between complexity and regular sphere packings."

G. Corso, "Small world effect in natural numbers network" (preprint 09/03)

[abstract:] "We develop a network in which the natural numbers are the vertices. We use the decomposition of natural numbers by prime numbers to establish the connections. We perform data collapse and show that the degree distribution of these networks scale linearly with the number of vertices. We compare the average distance of the network and the clustering coefficient with the distance and clustering coefficient of the corresponding random graph. In case we set connections among vertices each time the numbers share a common prime number the network is not a small-world type. If the criterium for establishing links becomes more selective, only prime numbers greater than pl are used to establish links, the network shows small-world effect, it means, it has high clustering coefficient and low distance."

G. Corso, "Families and clustering in a natural numbers network", Phys. Rev. E 69 (2004)

"We develop a network in which the natural numbers are the vertices. The decomposition of natural numbers by prime numbers is used to establish the connections. We perform data collapse and show that the degree distribution of these networks scales linearly with the number of vertices. We explore the families of vertices in connection with prime numbers decomposition. We compare the average distance of the network and the clustering coefficient with the distance and clustering coefficient of the corresponding random graph. In case we set connections among vertices each time the numbers share a common prime number the network has properties similar to a random graph. If the criterion for establishing links becomes more selective, only prime numbers greater than pl are used to establish links, where the network has high clustering coefficient."

A.C. Kumar and S. Dasgupta, "A small world network of prime numbers",   Physica A 357 (2005) 436

[abstract:] "According to Goldbach conjecture, any even number can be broken up as the sum of two prime numbers : $n = p + q$. We construct a network where each node is a prime number and corresponding to every even number $n$, we put a link between the component primes $p$ and $q$. In most cases, an even number can be broken up in many ways, and then we chose {\em one} decomposition with a probability $|p - q|^{\alpha}$. Through computation of average shortest distance and clustering coefficient, we conclude that for $\alpha > -1.8$ the network is of small world type and for $\alpha < -1.8$ it is of regular type. We also present a theoretical justification for such behaviour."

K.H. Knuth, "Measuring on lattices" (Presented at the 29th International Workshop on Bayesian and Maximum Entropy Methods in Science and Engineering: MaxEnt 2009)

[abstract:] "Previous derivations of the sum and product rules of probability theory relied on the algebraic properties of Boolean logic. Here they are derived within a more general framework based on lattice theory. The result is a new foundation of probability theory that encompasses and generalizes both the Cox and Kolmogorov formulations. In this picture probability is a bi-valuation defined on a lattice of statements that quantifies the degree to which one statement implies another. The sum rule is a constraint equation that ensures that valuations are assigned so as to not violate associativity of the lattice join and meet. The product rule is much more interesting in that there are actually two product rules: one is a constraint equation arises from associativity of the direct products of lattices, and the other a constraint equation derived from associativity of changes of context. The generality of this formalism enables one to derive the traditionally assumed condition of additivity in measure theory, as well introduce a general notion of product. To illustrate the generic utility of this novel lattice-theoretic foundation of measure, the sum and product rules are applied to number theory. Further application of these concepts to understand the foundation of quantum mechanics is described in a joint paper in this proceedings."

L. Alexandrov and L. Georgiev, "Prime number diffeomorphisms, Diophantine equations and the Riemann Hypothesis" (preprint 11/04)

[abstract:] "We explicitly construct a diffeomorphic pair (p(x),p-1(x)) in terms of an appropriate quadric spline interpolating the prime series. These continuously differentiable functions are the smooth analogs of the prime series and the prime counting function, respectively, and contain the basic information about the specific behavior of the primes. We employ p-1(x) to find approximate solutions of Diophantine equations over the primes and discuss how this function could eventually be used to analyze the von Koch estimate for the error in the prime number theorem which is known to be equivalent to the Riemann hypothesis."

J. Lagacé and L. Parovski, "A generalised Gauss circle problem and integrated density of states" (preprint 06/2015)

[abstract:] "Counting lattice points inside a ball of large radius in Euclidean space is a classical problem in analytic number theory, dating back to Gauss. We propose a variation on this problem: studying the asymptotics of the measure of an integer lattice of affine planes inside a ball. The first term is the volume of the ball; we study the size of the remainder term. While the classical problem is equivalent to counting eigenvalues of the Laplace operator on the torus, our variation corresponds to the integrated density of states of the Laplace operator on the product of a torus with Euclidean space. The asymptotics we obtain are then used to compute the density of states of the magnetic Schrödinger operator."

V. Balasubramanian, J.R. Fliss, R.G. Leigh and O. Parrikar, Multi-boundary entanglement in Chern–Simons theory and link invariants" (preprint 11/2016)

[abstract:] "We consider Chern–Simons theory for gauge group $G$ at level $k$ on 3-manifolds $M_n$ with boundary consisting of $n$ topologically linked tori. The Euclidean path integral on $M_n$ defines a quantum state on the boundary, in the $n$-fold tensor product of the torus Hilbert space. We focus on the case where $M_n$ is the link-complement of some $n$-component link inside the three-sphere $S^3$. The entanglement entropies of the resulting states define new, framing-independent link invariants which are sensitive to the topology of the chosen link. For the Abelian theory at level $k$ ($G= U(1)_k$) we give a general formula for the entanglement entropy associated to an arbitrary $(m|n-m)$ partition of a generic $n$-component link into sub-links. The formula involves the number of solutions to certain Diophantine equations with coefficients related to the Gauss linking numbers (mod $k$) between the two sublinks. This formula connects simple concepts in quantum information theory, knot theory, and number theory, and shows that entanglement entropy between sublinks vanishes if and only if they have zero Gauss linking (mod $k$). For $G = SU(2)_k$, we study various two and three component links. We show that the 2-component Hopf link is maximally entangled, and hence analogous to a Bell pair, and that the Whitehead link, which has zero Gauss linking, nevertheless has entanglement entropy. Finally, we show that the Borromean rings have a "W-like" entanglement structure (i.e., tracing out one torus does not lead to a separable state), and give examples of other 3-component links which have "GHZ-like" entanglement (i.e., tracing out one torus does lead to a separable state)."

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G. Chalmers, "Data compression with prime numbers" (preprint 11/05)

[abstract:] "A compression algorithm is presented that uses the set of prime numbers. Sequences of numbers are correlated with the prime numbers, and labeled with the integers. The algorithm can be iterated on data sets, generating factors of doubles on the compression."

[abstract:] "Riemann Hypothesis is viewed as a statement about the power dissipated in an electrical network."

P. Amore, "Convergence acceleration of series through a variational approach" (preprint 08/04)

[abstract:] "By means of a variational approach we find new series representations both for well known mathematical constants, such as and the Catalan constant, and for mathematical functions, such as the Riemann zeta function. The series that we have found are all exponentially convergent and provide quite useful analytical approximations. With limited effort our method can be applied to obtain similar exponentially convergent series for a large class of mathematical functions."

K. Maslanka, "Hypergeometric representation of the zeta-function of Riemann"
plus a few thoughts from a cosmologist on the nature of the zeta function

S. Tyagi and C. Holm, "A new integral representation for the Riemann Zeta function" (preprint 03/2007)

C. Krattenthaler, T. Rivoal, "On the integrality of the Taylor coefficients of mirror maps" (preprint 09/2007)

[abstract:] "We show that the Taylor coefficients of the series ${\bf q}(z)=z\exp({\bf G}(z)/{\bf F}(z))$ are integers, where ${\bf F}(z)$ and ${\bf G}(z)+\log(z) {\bf F}(z)$ are specific solutions of certain hypergeometric differential equations with maximal unipotent monodromy at $z=0$. We also address the question of finding the largest integer $u$ such that the Taylor coefficients of ${\bf q}(z)^{1/u}$ are still integers. As consequences, we are able to prove numerous integrality results for the Taylor coefficients of mirror maps of Calabi-Yau complete intersections in weighted projective spaces, which improve and refine previous results by Lian and Yau, and by Zudilin. In particular, we prove the general ''integrality'' conjecture of Zudilin about these mirror maps. A further outcome of the present study is the determination of the Dwork-Kontsevich sequence $(u_N)_{N\ge1}$, where $u_N$ is the largest integer such that $q(z)^{1/u_N}$ is a series with integer coefficients, where $q(z)=\exp(F(z)/G(z))$, $F(z)=\sum_{m=0} ^{\infty} (Nm)! z^m/m!^N$ and $G(z)=\sum_{m=1} ^{\infty} H_{Nm}(Nm)! z^m/m!^N$, with $H_n$ denoting the $n$-th harmonic number, conditional on the conjecture that there are no prime number $p$ and integer $N$ such that the $p$-adic valuation of $H_N$ is strictly greater than 3."

J.S. Brauchart, D.P. Hardin, E.B. Saff, "The Riesz energy of the $N$-th roots of unity: an asymptotic expansion for large $N$" (prepring 08/2008)

[abstract:] "We derive the complete asymptotic expansion in terms of powers of $N$ for the Riesz $s$-energy of $N$ equally spaced points on the unit circle as $N\to \infty$. For $s\ge -1$, such points form optimal energy $N$-point configurations with respect to the Riesz potential $1/r^{s}$, $s\neq0$, where $r$ is the Euclidean distance between points. By analytic continuation we deduce the expansion for all complex values of $s$. The Riemann zeta function plays an essential role in this asymptotic expansion."

J. Cano, M. Cheng, M. Mulligan, C. Nayak, E. Plamadeala and J. Yard, "Bulk-edge correspondence in $2+1$-dimensional Abelian topological phases" (preprint 10/2013)

[abstract:] "The same bulk two-dimensional topological phase can have multiple distinct, fully-chiral edge phases. We show that this can occur in the integer quantum Hall states at $\nu = 8$ and $12$, with experimentally-testable consequences. We show that this can occur in Abelian fractional quantum Hall states as well, with the simplest examples being at $\nu = 8/7, 12/11, 8/15, 16/5$. We give a general criterion for the existence of multiple distinct chiral edge phases for the same bulk phase and discuss experimental consequences. Edge phases correspond to lattices while bulk phases correspond to genera of lattices. Since there are typically multiple lattices in a genus, the bulk-edge correspondence is typically one-to-many; there are usually many stable fully chiral edge phases corresponding to the same bulk. We explain these correspondences using the theory of integral quadratic forms. We show that fermionic systems can have edge phases with only bosonic low-energy excitations and discuss a fermionic generalization of the relation between bulk topological spins and the central charge. The latter follows from our demonstration that every fermionic topological phase can be represented as a bosonic topological phase, together with some number of filled Landau levels. Our analysis shows that every Abelian topological phase can be decomposed into a tensor product of theories associated with prime numbers $p$ in which every quasiparticle has a topological spin that is a $p^n$-th root of unity for some $n$. It also leads to a simple demonstration that all Abelian topological phases can be represented by $U(1)^N$ Chern–Simons theory parameterized by a K-matrix."

M. Kibler, "Generalized spin bases for quantum chemistry and quantum information" (preprint 07/2008)

[abstract:] "Symmetry adapted bases in quantum chemistry and bases adapted to quantum information share a common characteristics: both of them are constructed from subspaces of the representation space of the group SO(3) or its double group (i.e., spinor group) SU(2). We exploit this fact for generating spin bases of relevance for quantum systems with cyclic symmetry and equally well for quantum information and quantum computation. Our approach is based on the use of generalized Pauli matrices arising from a polar decomposition of SU(2). This approach leads to a complete solution for the construction of mutually unbiased bases in the case where the dimension d of the considered Hilbert subspace is a prime number. We also give the starting point for studying the case where d is the power of a prime number. A connection of this work with the unitary group U(d) and the Pauli group is brielly underlined."

M.R. Kibler, "On mutually unbiased bases: Passing from $d$ to $d^2$" (preprint 10/2012)

[abstract:] "We show how to transform the problem of finding $d+1$ mutually unbiased bases in the $d$-dimensional Hilbert space into the one of finding $d(d+1)$ vectors in the $N$-dimensional Hilbert space with $N=d^2$. The transformation formulas admit a solution when $d$ is a prime number."

R. Olofsson, "Hecke eigenfunctions of quantized cat maps modulo prime powers", Annales Henri Poincaré 10 (6) (2009) 1111–1139

[abstract:] "This paper continues the work done in Olofsson [Commun. Math. Phys. 286(3):1051–1072, 2009] about the supremum norm of eigenfunctions of desymmetrized quantized cat maps. $N$ will denote the inverse of Planck's constant and we will see that the arithmetic properties of $N$ play an important role. We prove the sharp estimate $||\psi ||_{\infty} = O(N^{1/4})$ for all normalized eigenfunctions and all $N$ outside of a small exceptional set. We are also able to calculate the value of the supremum norms for most of the so called newforms. For a given $N = p^n$ , with $n>2$, the newforms can be divided in two parts (leaving out a small number of them in some cases), the first half all have supremum norm about $2/\sqrt{1\pm 1/p}$ and the supremum norm of the newforms in the second half have at most three different values, all of the order $N^1/6$. The only dependence of $A$ is that the normalization factor is different if $A$ has eigenvectors modulo $p$ or not. We also calculate the joint value distribution of the absolute value of $n$ different newforms."

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

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

W.T. Buttler, S.K. Lamoreaux and J.R. Torgerson, "A four-level single-photon quantum cryptography system based on polarization, phase and time encoding" (preprint 03/2010)

[abstract:] "We describe a quantum cryptography protocol with up to twelve four-dimensional ($d = 4$) states generated by a polarization-, phase- and time-encoding transmitter. This protocol can be experimentally realized with existing technology, drawing from time-encoded and polarization-encoded systems. The protocol is error tolerant and has a quantum bit-rate of 2 per transmission, which when combined with state detection efficiency yields a qubit efficiency of up to 1 or double that of BB84-{\it like} protocols. As a practical system, our result appears to contradict a fundamental theorem stating that there exists $d + 1$ maximally non-orthogonal bases for a $d$-dimensional space where $d$ is the power of a prime number. Evidently, this contradiction has its origin in the difference in the size of the vector spaces spanned by the basis states, semi-infinite in time and phase in our case, vs. a finite number of polarization states alone as previously considered."

S. Foldes, "The Lorentz group and its finite field analogues: local isomorphism and approximation" (preprint 05/2008)

[abstract:] "Finite Lorentz groups acting on 4-dimensional vector spaces coordinatized by finite fields with a prime number of elements are represented as homomorphic images of countable, rational subgroups of the Lorentz group acting on real 4-dimensional space-time. Bounded subsets of the real Lorentz group are retractable with arbitrary accuracy to finite subsets of such rational subgroups. These finite retracts correspond, via local isomorphisms, to well-behaved subsets of Lorentz groups over finite fields. This establishes a relationship of approximation between the real Lorentz group and Lorentz groups over very large finite fields."

M. Fujimoto, K. Uehara, "Regularizations of the Euler product representation for zeta functions and the Birch-Swinnerton-Dyer conjecture" (preprint 09/2007)

[abstract:] "We consider a variant expression to regularize the Euler product representation of the zeta functions, where we mainly apply to that of the Riemann zeta function in this paper. The regularization itself is identical to that of the zeta function of the summation expression, but the non-use of the Möbius function enable us to confirm a finite behavior of residual terms which means an absence of zeros except for the critical line. Same technique can be applied to the L-function associated to the elliptic curve, and we can deal with the Taylor expansion at the pole in critical strip which is deeply related to the Birch-Swinnerton-Dyer conjecture."

M. Fujimoto and K. Uehara, "Regularized Euler product for the zeta function and the Birch and Swinnerton-Dyer and the Beilinson conjecture" (preprint 11/2008)

[Abstract:] "We present another expression to regularize the Euler product representation of the Riemann zeta function. In this paper. The expression itself is essentially same as the usual Euler product that is the infinite product, but we define a new one as the limit of the product of some terms derived from the usual Euler product. We also refer to the relation between the Bernoulli number and $P(z)$, which is an infinite summation of a $z$ power of the inverse primes. When we apply the same technique to the $L$-function associated to an elliptic curve, we can evaluate the power of the Taylor expansion for the function even in the critical strip, which is deeply related to problems known as the Birch and Swinnerton-Dyer conjecture and the Beilinson conjecture."

M.W. Coffey and M.C. Lettington, "Mellin transforms with only critical zeros: Chebyshev and Gegenbauer functions" (preprint 07/2013)

[abstract:] "We consider the Mellin transforms of certain Chebyshev functions based upon the Chebyshev polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line or on the real line. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. Furthermore, we extend this result to a 1-parameter family of polynomials with zeros only on the critical line. These polynomials possess the functional equation $p_n(s;\beta)=(-1)^{\lfloor n/2 \rfloor} p_n (1-s;\beta)$. We then present the generalization to the Mellin transform of certain Gegenbauer functions. The results should be of interest to special function theory, combinatorics, and analytic number theory."

M.W. Coffey and M.C. Lettington, "Mellin transforms with only critical zeros: Legendre functions" (preprint 07/2013)

[abstract:] "We consider the Mellin transforms of certain Legendre functions based upon the ordinary and associated Legendre polynomials. We show that the transforms have polynomial factors whose zeros lie all on the critical line Re $s=1/2$. The polynomials with zeros only on the critical line are identified in terms of certain $_3F_2(1)$ hypergeometric functions. These polynomials possess the functional equation $p_n(s)= (-1)^{\lfloor n/2 \rfloor} p_n(1-s)$. Other hypergeometric representations are presented, as well as certain Mellin transforms of fractional part and fractional part-integer part functions. The results should be of interest to special function theory, combinatorial geometry, and analytic number theory."

M.W. Coffey, "Series representation of the Riemann zeta function and other results: Complements to a paper of Crandall" (preprint 03/2012)

[abstract:] "We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions providing analytic continuation through out the whole complex plane. Additionally we demonstrate some series representations for the initial Stieltjes constants appearing in the Laurent expansion of the Hurwitz zeta function. A particular point of elaboration in these developments is the hypergeometric form and its equivalents for certain derivatives of the incomplete Gamma function. Finally, we evaluate certain integrals including $\int_{\tiny{Re} s=c} {{\zeta(s)} \over s} ds$ and $\int_{\tiny{Re} s=c} {{\eta(s)} \over s} ds$, with $\zeta$ the Riemann zeta function and $\eta$ its alternating form."

"We develop series representations for the Hurwitz and Riemann zeta functions in terms of generalized Bernoulli numbers (N\"{o}rlund polynomials), that give the analytic continuation of these functions to the entire complex plane. Special cases yield series representations of a wide variety of special functions and numbers, including log Gamma, the digamma, and polygamma functions. A further byproduct is that $\zeta(n)$ values emerge as nonlinear Euler sums in terms of generalized harmonic numbers. We additionally obtain series and integral representations of the first Stieltjes constant $\gamma_1(a)$. The presentation unifies some earlier results."

M.W. Coffey, "Evaluation of some second moment and other integrals for the Riemann, Hurwitz, and Lerch zeta functions" (preprint 02/2011)

[abstract:] "Several second moment and other integral evaluations for the Riemann zeta function $\zeta(s)$, Hurwitz zeta function $\zeta(s,a)$, and Lerch zeta function $\Phi(z,s,a)$ are presented. Additional corollaries that are obtained include previously known special cases for the Riemann zeta function $\zeta(s)=\zeta(s,1)=\Phi(1,s,1)$. An example special case is: $$\int_R {{|\zeta(1/2+it)|^2} \over {t^2+1/4}}dt=2\pi[\ln(2\pi)-\gamma],$$ with $\gamma$ the Euler constant. The asymptotic forms of certain fractional part integrals, with and without logarithmic factors in the integrand, are presented. Extensions and other approaches are mentioned."

M.W. Coffey, "Sums of alternating products of Riemann zeta values and solution of a Monthly problem" (preprint, 06/2011)

[abstract:] "We solve problem 11585 proposed by B. Burdick, AMM June-July 2011 {\bf 118} (6), p. 558 for the sum of certain products of Riemann zeta function values. We further point out an alternating sum analog, and then present and prove different alternating sum analogs. In addition, we present summation by parts and other results for the Hurwitz and Riemann zeta functions and for the digamma and trigamma functions."

M.W. Coffey, "Series of zeta values, the Stieltjes constants, and a sum S_\gamma(n)" (preprint 06/2007)

[abstract:] "We present a variety of series representations of the Stieltjes and related constants, the Stieltjes constants being the coefficients of the Laurent expansion of the Hurwitz zeta function zeta(s,a) about s=1. Additionally we obtain series and integral representations of a sum S_\gamma(n) formed as an alternating binomial series from the Stieltjes constants. The slowly varying sum S_\gamma(n)+n is an important subsum in application of the Li criterion for the Riemann hypothesis."

M.W. Coffey, "The Stieltjes constants, their relation to the eta_j coefficients, and representation of the Hurwitz zeta function" (preprint 06/2007)

[abstract:] "The Stieltjes constants gamma_k(a) are the expansion coefficients in the Laurent series for the Hurwitz zeta function about its only pole at s=1. We present the relation of gamma_k(1) to the eta_j coefficients that appear in the Laurent expansion of the logarithmic derivative of the Riemann zeta function about its pole at s=1. We obtain novel integral representations of the Stieltjes constants and new decompositions such as S_2(n) = S_gamma(n) + S_Lambda(n) for the crucial oscillatory subsum of the Li criterion for the Riemann hypothesis. The sum S_\gamma(n) is O(n) and we present various integral representations for it. We present novel series representations of S_2(n). We additionally present a rapidly convergent expression for \gamma_k= \gamma_k(1) and a variety of results pertinent to a parameterized representation of the Riemann and Hurwitz zeta functions."

Mark W. Coffey, "On representations and differences of Stieltjes coefficients, and other relations" (preprint 09/2008)

[abstract:] "The Stieltjes coefficients $\gamma_k(a)$ arise in the expansion of the Hurwitz zeta function $\zeta(s,a)$ about its single simple pole at $s=1$ and are of fundamental and long-standing importance in analytic number theory and other disciplines. We present an array of exact results for the Stieltjes coefficients, including series representations and summatory relations. Other integral representations provide the difference of Stieltjes coefficients at rational arguments. The presentation serves to link a variety of topics in analysis and special function and special number theory, including logarithmic series, integrals, and the derivatives of the Hurwitz zeta and Dirichlet $L$-functions at special points. The results have a wide range of application, both theoretical and computational."

M. Coffey, "On the coefficients of the Baez-Duarte criterion for the Riemann hypothesis and their extensions" (preprint 08/2006)

[abstract:] "We discuss analytic properties of the constants ck appearing in the Baez-Duarte criterion for the Riemann hypothesis. These constants are the coefficients of Pochhammer polynomials in a series representation of the reciprocal of the Riemann zeta function. We present extensions of this representation to the Hurwitz zeta and many other special functions. We relate the corresponding coefficients to other known constants including the Stieltjes constants and present summatory relations. In addition, we generalize the Maslanka hypergeometric-like representation for the zeta function in several ways."

M.W. Coffey, "Special functions and the Mellin transforms of Laguerre and Hermite functions" (preprint 12/2006)

[abstract:] "We present explicit expressions for the Mellin transforms of Laguerre and Hermite functions in terms of a variety of special functions. We show that many of the properties of the resulting functions, including functional equations and reciprocity laws, are direct consequences of transformation formulae of hypergeometric functions. Interest in these results is reinforced by the fact that polynomial or other factors of the Mellin transforms have zeros only on the critical line Re s = 1/2. We additionally present a simple-zero Proposition for the Mellin transform of the wavefunction of the D-dimensional hydrogenic atom. These results are of interest to several areas including quantum mechanics and analytic number theory."

M.W. Coffey, "On harmonic binomial series" (preprint 12/2008)

[abstract:] "We evaluate binomial series with harmonic number coefficients, providing recursion relations, integral representations, and several examples. The results are of interest to analytic number theory, the analysis of algorithms, and calculations of theoretical physics, as well as other applications."

M.W. Coffey, "Some definite logarithmic integrals from Euler sums, and other integration results" (preprint 01/2010)

[abstract:] "We present explicit expressions for multi-fold logarithmic integrals that are equivalent to sums over polygamma functions at integer argument. Such relations find application in perturbative quantum field theory, quantum chemistry, analytic number theory, and elsewhere. The analysis includes the use of properties of a variety of special functions."

S. Albino, "Analytic Continuation of Harmonic Sums" (preprint 02/2009)

[abstract:] "We present a method for calculating any harmonic sum to arbitrary accuracy for all complex values of the argument. The method utilizes the relation between harmonic sums and (derivatives of) generalized Riemann zeta functions, which allows a harmonic sum to be calculated as a Taylor series in the inverse of its argument. A program for implementing this method is also provided."

[abstract:] "In this article we provide integral representations for Dirichlet beta and Riemann zeta functions, which are obtained by combining Mellin transform with fractional cosine (sine) transform. As an application of these integral formulas we derive expansions of the above L-functions in the series of Meixner-Pollaczek polynomials and Pochhammer functions."

M. Combescure, "The Mutually Unbiased Bases revisited" (preprint 05/2006)

[abstract:] "The study of Mutually Unbiased Bases continues to be developed vigorously, and presents several challenges in the Quantum Information Theory. Two orthonormal bases in $\mathbb C^d, B {and} B'$ are said mutually unbiased if $\forall b\in B, b'\in B'$ the scalar product $b\cdot b'$ has modulus $d^{-1/2}$. In particular this property has been introduced in order to allow an optimization of the measurement-driven quantum evolution process of any state $\psi \in \mathbb C^d$ when measured in the mutually unbiased bases $B\_{j} {of} \mathbb C^d$. At present it is an open problem to find the maximal number of mutually Unbiased Bases when $d$ is not a power of a prime number.In this article, we revisit the problem of finding Mutually Unbiased Bases (MUB's) in any dimension $d$. The method is very elementary, using the simple unitary matrices introduced by Schwinger in 1960, together with their diagonalizations. The Vandermonde matrix based on the $d$-th roots of unity plays a major role. This allows us to show the existence of a set of 3 MUB's in any dimension, to give conditions for existence of more than 3 MUB's for $d$ even or odd number, and to recover the known result of existence of $d+1$ MUB's for $d$ a prime number. Furthermore the construction of these MUB's is very explicit. As a by-product, we recover results about Gauss Sums, known in number theory, but which have apparently not been previously derived from MUB properties."

C. Spengler and B. Kraus, "A graph state formalism for mutually unbiased bases" (preprint 09/2013)

[abstract:] "A pair of orthonormal bases is called mutually unbiased if all mutual overlaps between any element of one basis with an arbitrary element of the other basis coincide. In case the dimension, $d$, of the considered Hilbert space is a power of a prime number, complete sets of $d+1$ Mutually Unbiased Bases (MUBs) exist. Here, we present a novel method, which uses the graph state formalism to construct MUBs. We show, that for $n$ $p$-level systems, with $p$ being prime, one generalized graph state suffices to easily construct the corresponding complete set of $p^n+1$ MUBs. In fact, we show that a single $n$-dimensional vector, which is associated to the generalized graph state, can be used to generate the complete set of MUBs and demonstrate that these vectors can be easily determined."

C. Nash and D. O'Connor, "Ray-Singer torsion, topological field theories and the Riemann zeta function at s = 3"

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."

S. Torquato and Y. Jiao, "Dense packings of the Platonic and Archimedean solids", Nature 460 (2009) 876-879

[abstract:] "Dense packings have served as useful models of the structure of liquid, glassy and crystal states of matter, granular media, heterogeneous materials, and biological systems. Probing the symmetries and other mathematical properties of the densest packings is a problem of long-standing interest in discrete geometry and number theory. The preponderance of previous work has focused on spherical particles, and very little is known about dense polyhedral packings. We formulate the problem of generating dense packings of polyhedra within an adaptive fundamental cell subject to periodic boundary conditions as an optimization problem, which we call the Adaptive Shrinking Cell (ASC) scheme. This novel optimization problem is solved here (using a variety of multi-particle initial configurations) to find dense packings of each of the Platonic solids in three-dimensional Euclidean space. We find the densest known packings of tetrahedra, octahedra, dodecahedra and icosahedra with densities $0.782...$, $0.947...$, $0.904...$, and $0.836...$, respectively. Unlike the densest tetrahedral packing, which must be a non-Bravais lattice packing, the densest packings of the other non-tiling Platonic solids that we obtain are their previously known optimal (Bravais) lattice packings. Our simulations results, rigorous upper bounds that we derive, and theoretical arguments lead us to the strong conjecture that the densest packings of the Platonic and Archimedean solids with central symmetry are given by their corresponding densest lattice packings. This is the analog of Kepler's sphere conjecture for these solids."

S. Haran, "Index theory, potential theory, and the Riemann Hypothesis",   L-functions and Arithmetic, Durham 1990, LMS Lecture Notes   153 (1991), 257-270.

I. Fesenko, "Several nonstandard remarks" (preprint, 2003)

[abstract:] "This text aims to present and discuss a number of situations in analysis, geometry, number theory and mathematical physics which can profit from developing their nonstandard description or interpretation and then using it to prove standard results and/or establish standard theories."

Section 8 concerns "Nonstandard interpretations of interactions between noncommutative differential geometry and number theory."

[excerpt p.8-9:]

M. du Sautoy, J. McDermott, and G. Smith, "Zeta functions of crystallographic groups and analytic continuation", Proceedings of the London Mathematical Society (3) 79 (1999) 511-534.

D. Bessis, J. Geronimo, and P. Moussa, "Mellin transforms associated with Julia sets and physical applications", Journal of Statistical Physics 34 (1984) 75-110

Xu Xinwen and Dai Xianxi, "On a specific heat-phonon spectrum inversion problem: Exact solution, unique existence theorem and Riemann Hypothesis", Physics Letters A 147 8,9 (1990) 445-449.

S. Garoufalidis, P. Kucharski and P. Sulkowski, "Knots, BPS states, and algebraic curves" (preprint 04/2015)

"We analyze relations between BPS degeneracies related to Labastida–Marino–Ooguri–Vafa (LMOV) invariants, and algebraic curves associated to knots. We introduce a new class of such curves that we call extremal A-polynomials, discuss their special properties, and determine exact and asymptotic formulas for the corresponding (extremal) BPS degeneracies. These formulas lead to nontrivial integrality statements in number theory, as well as to an improved integrality conjecture stronger than the known M-theory integrality predictions. Furthermore we determine the BPS degeneracies encoded in augmentation polynomials and show their consistency with known colored HOMFLY polynomials. Finally we consider refined BPS degeneracies for knots, determine them from the knowledge of super-A-polynomials, and verify their integrality. We illustrate our results with twist knots, torus knots, and various other knots with up to 10 crossings."

Y. Kano and E. Wolf, "Temporal coherence of black body radiation", Proc. Phys. Soc. 80 (1962) 1273-1276

[abstract:] "The temporal complex coherence function of black-body radiation is calculated and is found to be expressible in terms of the generalized Riemann $\xi$-function. Curves are given which show the variation of the absolute value and of the argument of the temporal complex degree of coherence $\gamma(\tau)$ as functions of increasing time delay $\tau$. It is shown that the analytic continuation of $\gamma(\tau)$ has no zeros in the lower half of the complex plane. This result supports the theory proposed in the accompanying paper by Wolf about the possibility of determining certain energy spectra from measurements of the absolute value of the degree of coherence."

R. Marrett, "Aggregate properties of fracture populations", Journal of Structural Geology 18 (1996) 169-178

[abstract:] "Empirical studies indicate that the individual attributes of both faults and extension fractures follow power-law scaling. Aggregate properties of fracture populations are important in a variety of problems and can be specified in terms of the scaling parameters of individual fracture attributes. Development of an expression for an aggregate property requires consideration of a number of independent factors, including the topologic dimension of the aggregate property, the topologic dimension of sampling and the possibility of scaling changes for fractures that span some mechanically significant layer. The Riemann zeta function provides an alternative to integration for the analytical and numerical solution of aggregate problems."

[personal communication from author, 11/07/03:] "See section on "Evaluation," pages 173-174, for the part that might be most germane to your interests. My usage of the Riemann zeta function is shallow, really only to deal with an infinite power series. In geology this traditionally has been done (poorly) using integration, which I show to systematically underestimate the aggregate effects of individual features that follow a power law distibution of sizes.   I have not seen this aspect of my paper used in later publications (geologists are notoriously weak in numerical and analytical skills). Likewise, I have not seen the Riemann zeta function used in other geological contexts; I stumbled on it in my physics and math books while searching for a better approach to summation of infinite power series."

S. Benvenuti, B. Feng, A. Hanany, Yang-Hui He, "BPS operators in gauge theories: Quivers, syzygies and plethystics" (preprint 08/2006)

[abstract:] "We develop a systematic and efficient method of counting single-trace and multi-trace BPS operators for world-volume gauge theories of N D-brane probes, for both N -> infinity and finite N. The techniques are applicable to generic singularities, orbifold, toric, non-toric, et cetera, even to geometries whose precise field theory duals are not yet known. Mathematically, fascinating and intricate inter-relations between gauge theory, algebraic geometry, combinatorics and number theory exhibit themselves in the form of plethystics and syzygies."

M. Hippke, et al., "Pulsation period variations in the RRc Lyrae star KIC 5520878" (preprint 09/2014) (Submitted on 3 Sep 2014)

"Learned et al. proposed that a sufficiently advanced extra-terrestrial civilization may tickle Cepheid and RR Lyrae variable stars with a neutrino beam at the right time, thus causing them to trigger early and jogging the otherwise very regular phase of their expansion and contraction. This would turn these stars into beacons to transmit information throughout the galaxy and beyond. The idea is to search for signs of phase modulation (in the regime of short pulse duration) and patterns, which could be indicative of intentional, omnidirectional signaling.

We have performed such a search among variable stars using photometric data from the Kepler space telescope. In the RRc Lyrae star KIC 5520878, we have found two such regimes of long and short pulse durations. The sequence of period lengths, expressed as time series data, is strongly auto correlated, with correlation coefficients of prime numbers being significantly higher (p = 99.8%). Our analysis of this candidate star shows that the prime number oddity originates from two simultaneous pulsation periods and is likely of natural origin.

Simple physical models elucidate the frequency content and asymmetries of the KIC 5520878 light curve.

Despite this SETI null result, we encourage testing other archival and future time-series photometry for signs of modulated stars. This can be done as a by-product to the standard analysis, and even partly automated."

L.B. Garrido, J. Gainza and E. Pereira, "Rheological properties of concentrated aqueous suspensions of kaolin - application of the multiple ternary Bingham model", Applied Clay Science 5 (1990) 217-228

[abstract:] "Several kaolin samples from Monte Salgueiro (Spain) obtained by hydrocycloning, wet ball milling and delamination were examined. The specific lateral and basal surface areas of the kaolinite were calculated from both the specific surface area by the BET method and the particle size distribution curves. Flow curves of aqueous suspensions (51% w/w) of those kaolins were obtained by a rotational viscometer of coaxial cylinders with a predetermined history of flow. To describe the rheological behaviour the mechanical ternary Bingham model was applied satisfactorily. Its differential equation under the movement conditions has been integrated according to the flow history selected. The model and consequently the resulting equation have been generalized linearly. The number of rheological parameters was reduced to three because of the application of empirical relations among them. These parameters were ordered according to the Riemann zeta function simplifying the calculation. Rheological changes produced by mechanical processes can be described using the parameters of the model which have been correlated with the geometric mean diameter, specific surface area and cationic exchange capacity by multiple regression."

P. Exner, P. Seba, "A Markov process associated with plot-size distribution in Czech Land Registry and its number-theoretic properties" (preprint 12/07)

[abstract:] "The size distribution of land plots is a result of land allocation processes in the past. In the absence of regulation this is a Markov process leading an equilibrium described by a probabilistic equation used commonly in the insurance and financial mathematics. We support this claim by analyzing the distribution of two plot types, garden and build-up areas, in the Czech Land Registry pointing out the coincidence with the distribution of prime number factors described by Dickman function in the first case."

"The motion in the complex plane of the zeros to various zeta functions is investigated numerically. First the Hurwitz zeta function is considered and an accurate formula for the distribution of its zeros is suggested. Then functions which are linear combinations of different Hurwitz zeta functions, and have a symmetric distribution of their zeros with respect to the critical line, are examined. Finally the existence of the hypothetical non-trivial Riemann zeros with Re(s) /neq 1/2 is discussed."

L. Alexandrov, "On the nonasymptotic prime number distribution"

[abstract:] "The objective of this paper is to introduce an approach to the study of the nonasymptotic distribution of prime numbers. The natural numbers are represented by theorem 1 in the matrix form ^2N. The first column of the infinite matrix ^2N starts with the unit and contains all composite numbers in ascending order.The infinite rows of this matrix except for the first elements contain prime numbers only, which are determined by an uniform recurrence law. At least one of the elements of the twin pairs of prime numbers is an element of the second column of the matrix ^2N (theorem 3). The basic information on the nonasymptotic prime number distribution is contained in the distribution of the elements of the second column of the matrix ^2N."

[abstract:] "Due to the sieving process represented by a Secondary Sieving Map; during the generation of the prime numbers, geometric structures with definite symmetries are formed which become evident through their geometrical representations. The study of these structures allows the development of a constructive prime generating formula. This defines a mean prime density yielding a second order recursive and discrete prime producing formula and a second order differential equation whose solutions produce an improved Prime Number Theorem. Applying these results to twin prime pairs is possible to generate a Twin Prime Number Theorem and important conclusions about the infinitude of the twin primes."

J.S. Dowker and K. Kirsten, "The Barnes zeta-function, sphere determinants and Glaisher- Kinkelin-Bendersky constants"

[abstract:] "Summations and relations involving the Hurwitz and Riemann zeta-functions are extended first to Barnes zeta-functions and then to zeta-functions of general type. The analysis is motivated by the evaluation of determinants on spheres which are treated both by a direct expansion method and by regularised sums. Comments on existing calculations are made. A Kaluza-Klein technique is introduced providing a determinant interpretation of the Glaisher-Kinkelin-Bendersky constants which are then generalised to arbitrary zeta-functions. This technique allows an improved treatment of sphere determinants."

A. Al-Shuaibi,   "The Riemann zeta function used in the inversion of the Laplace transform",   Inverse Problems 14 (1998) 1-7.

[Abstract:] "Given the Laplace transform F(s) of a function f(t), we develop a new algorithm to find an approximation to f(t) by the use of the classical Laguerre polynomials. The main contribution of our work is the development of a new and very effective method to evaluate the Laguerre coefficients with the use of the Riemann zeta function. Some examples are illustrated."

D. Bryukhov, "Axially symmetric generalization of the Cauchy-Riemann system and modified Clifford analysis" (preprint 02/03)

[abstract:] "The main aim of this paper is to describe the most adequate generalization of the Cauchy-Riemann system fixing properties of classical functions in octonionic case. An octonionic generalization of the Laplace transform is introduced. Octonionic generalizations of the inversion transformation, the gamma function and the Riemann zeta-function are given."

N.V. Kuznetsov, "The true order of the Riemann zeta-function"

"For the Riemann zeta function on the critical line the terminal estimate has been proved, which had been conjectured by Lindelof at the beginning of this century. The proof is based on the author's relations which connect the bilinear forms of the eigenvalues of the Hecke operators with sums of the Kloosterman sums."

M. Wolf, "Nearest neighbor spacing distribution of prime numbers and quantum chaos" (preprint 12/2012)

[abstract:] "We show that after appropriate rescaling the statistics of spacings between adjacent prime numbers follows the Poisson distribution. The scaling transformation removes the oscillations in the NNSD of primes. These oscillations have the very profound period of length six. We calculate the spectral rigidity $\Delta_3$ for prime numbers by two methods and we find the cross-overs in their behaviors."

G. G. Szpiro, "The gaps between the gaps: some patterns in the prime number sequence", Physica A 341 (2004) 607-617

[abstract:] "It has long been known that the gaps between consecutive prime numbers cluster on multiples of 6. Recently it was shown that the frequency of the gaps between the gaps is lower for multiples of 6 than for other values (P. Kumar et. al., "Information entropy and correlation in prime numbers"). This paper investigates "higher moments" of the prime number series and shows that they exhibit certain peculiarities. In order to remove doubts as to whether these peculiarities are related to the known clustering of the gaps on multiples of 6, the results are compared to a benchmark series of "simulated gaps"."

C. Feinstein, "Complexity theory for simpletons"     (preprint, 07/2005)

[abstract:] "In this article, we shall describe some of the most interesting topics in the subject of Complexity Theory for a general audience. Anyone with a solid foundation in high school mathematics (with some calculus) and an elementary understanding of computer programming will be able to follow this article. First, we shall describe the P versus NP problem and its significance. Next, we shall describe two other famous mathematics problems, the Collatz 3n+1 Conjecture and the Riemann Hypothesis, and show how the notion of "computational irreducibility" is important for understanding why no one has, as of yet, solved these two problems."

M. Rubinstein, "Computational methods and experiments in analytic number theory" (preprint 12/04)

[abstract:] "We cover some useful techniques in computational aspects of analytic number theory, with specific emphasis on ideas relevant to the evaluation of L-functions. These techniques overlap considerably with basic methods from analytic number theory. On the elementary side, summation by parts, Euler-Maclaurin summation, and Möbius inversion play a prominent role. In the slightly less elementary sphere, we find tools from analysis, such as Poisson summation, generating function methods, Cauchy's residue theorem, asymptotic methods, and the fast Fourier transform. We then describe conjectures and experiments that connect number theory and random matrix theory."

P. Kurlberg and Z. Rudnick, "The distribution of spacings between quadratic residues"

"We study the distribution of spacings between squares modulo q, where q is square-free and highly composite, in the limit as the number of prime factors of q goes to infinity. We show that all correlation functions are Poissonian which among other things, implies that the spacings between nearest neighbors, normalized to have unit mean, have an exponential distribution."

"In these lectures three different methods of computing the asymptotic expansion of a Hermitian matrix integral is presented... The second method is based on the classical analysis of orthogonal polynomials. A rigorous asymptotic method is established, and a special case of the matrix integral is computed in terms of the Riemann zeta-function."

M.K.-H. Kiessling, "Order and chaos in some trigonometric series" (preprint 06/2012)

[abstract:] "The one-parameter family of deterministic trigonometric series $\pzcS_p: t\mapsto \sum_{n\in\Nset}\sin(n^{-{p}}t)$, $p>1$, is shown to exhibit both order and apparent chaos. It is proved that $\pzcS_p(t) = \alpha_p\rm{sign}(t)|t|^{1/{p}}+O(|t|^{1/{(p+1)}})\;\forall\;t\in\Rset$, with explicitly computed constant $\alpha_p$. A well-motivated conjecture is formulated concerning the seemingly chaotic fluctuations about this overall trend, to the effect that these fluctuations, when properly scaled, converge in distribution to a standard Gaussian when $t\to\infty$, provided that $p$ is irrational; no conjecture has been forthcoming for rational $p$. Moreover, the interesting relationship of the asymptotics of $\pzcS_p(t)$ to properties of the Riemann $\zeta$ function is worked out."

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."

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."

P. Flajolet and I. Vardi, "Zeta function expansions of classical constants" (preprint, 1996)

"Langlands on Langlands" and "More On Geometric Langlands (a Grand Unified Theory of Math?)" (articles from Peter Woit's blog Not Even Wrong)

Workgroup: Zeta functions and locally symmetric spaces
(Technical University, Clausthal, Germany)

• Harmonic analysis and zeta functions
• Thermodynamic formalisms and dynamic zeta functions
• Arithmetic theory of zeta functions

"The model concepts enable to show that the continuum real number field contains unique structures, namely prime numbers which are analogous to the dominant eddies in the eddy continuum in turbulent fluid flows. In this paper it is shown that the prime number frequency spectrum follows quantumlike mechanical laws."

This is an elaboration of the following, earlier version:

"The quantum-like chaos in atmospheric flows can be quantified in terms of the following mathematical functions/concepts: (1) The fractal structure of the flow pattern is resolved into an overall logarithmic spiral trajectory with the quasiperiodic Penrose tiling pattern for the internal structure and is equivalent to a hierarchy of vortices. The incorporation of Fibonacci mathematical series, representative of ramified bifurcations, indicates ordered growth of fractal patterns. (2) The steady state emergence of progressively larger fractal structures incorporates unique primary perturbation domains of progressively increasing number equal to z/ln z where z, the length step growth stage is equal to the length scale ratio of large eddy to turbulent eddy. In number theory, z/ln z gives the number of primes less than     z. The model also predicts that z/ln z represents the normalised cumulative variance spectrum of the eddies and which follows statistical normal distribution. The important result of the study is that the prime number spectrum is the same as the eddy energy spectrum for quantum-like chaos in atmospheric flows."

The following article is unrelated to number theory, but provided a conceptual framework around which the above articles were constructed.

Gary W. Adamson has supplied four new entries to   Sloane's Encyclopedia of Integer Sequences, relating to the "heights" of the Riemann zeros. Access the search field at the website using "sequence number", and the following: A100060, A102522, A102523, A102524.

## books

L. Nyssen (ed.), Physics and Number Theory, IRMA Lectures in Mathematics and Theoretical Physics, vol. 10 (European Mathematics Society, 2006)

[publisher's description:] "There is a rich and historical relationship between theoretical physics and number theory. This volume presents a selection of problems which are currently in full development and inspire a lot of research going on. Each of the seven contributions starts with an introductory survey which makes it possible even for non-specialists to understand the results and to gain an idea of the great variety of subjects and techniques used.

Topics covered are: phase locking in oscillating systems, crystallography, Hopf algebras and renormalisation theory, Zeta-function and random matrices, Kloosterman sums and the local Langlands correspondence.

Intended for research mathematicians and theoretical physicists as well as graduate students, this volume gives an overview of recent developments in an exciting subject crossing several disciplines."

P.E. Cartier, B. Julia, P. Moussa and P. Vanhove (eds.), Frontiers in Number Theory, Physics, and Geometry: On Random Matrices, Zeta Functions, and Dynamical Systems (Springer, due March 2006)

[publisher's description:] "This book presents pedagogical contributions on selected topics relating Number Theory, Theoretical Physics and Geometry. The parts are composed of long self-contained pedagogical lectures followed by shorter contributions on specific subjects organized by theme. Most courses and short contributions go up to the recent developments in the fields; some of them follow their author's original viewpoints. There are contributions on Random Matrix Theory, Quantum Chaos, Non-commutative Geometry, Zeta functions, and Dynamical Systems. The chapters of this book are extended versions of lectures given at a meeting entitled Number Theory, Physics and Geometry, held at Les Houches in March 2003, which gathered mathematicians and physicists."

Number Theory and Physics, M. Waldschmidt, et al., eds. Proceedings of the Winter School, Les Houches, France, March 7–16, 1989, Springer Proceedings in Physics 47 (Springer, 1989)

From Number Theory to Physics, M. Waldschmidt, et al., eds. Further proceedings of the winter school, Les Houches, France, March 7–16, 1989 (Springer, 1992)

S.A. Burr, G.E. Andrews (editors), The Unreasonable Effectiveness of Number Theory (Proceedings of Symposia in Applied Mathematics, Vol. 46) (AMS, 1992).

D. Hejhal, et. al., eds., Emerging Applications of Number Theory, (Springer, 1999)

J.C.A. Boeyens, Demetrius C. Levendis, Number Theory and the Periodicity of Matter   (Springer, 2008)

[publisher's description:] "Philosophers have long speculated that a link exists between natural numbers and the physical world:

Pythagoras "all is number"
John Dalton (1803) "Atoms combine in simple numerical ratios"
Prout's hypothesis, published anonymously in 1815 "..the elements are different aggregates of the atoms of primordial hydrogen.."
Alexandre É Beguyer de Chancourtois (1862) "..the properties of the elements are the properties of numbers.."
William Harkins (1921) "the ratio Z/(A - Z) never exceeds the value 0.62 in atomic species"

Within this book, readers with an interest in mathematics, science or natural philosophy will find this expectation addressed."

Y. Motohashi, Spectral Theory of the Riemann Zeta-Function, (Cambridge University Press, 1999)

Spectral Analysis in Geometry and Number Theory, edited by M. Kotani and H. Naito (AMS, 2009)

[publisher's description:] "This volume is an outgrowth of an international conference in honor of Toshikazu Sunada on the occasion of his sixtieth birthday. The conference took place at Nagoya University, Japan, in 2007.

Sunada's research covers a wide spectrum of spectral analysis, including interactions among geometry, number theory, dynamical systems, probability theory and mathematical physics. Readers will find papers on trace formulae, isospectral problems, zeta functions, quantum ergodicity, random waves, discrete geometric analysis, value distribution, and semiclassical analysis. This volume also contains an article that presents an overview of Sunada's work in mathematics up to the age of sixty."

R. Crandall and C. Pomerance, Prime Numbers: A Computational Perspective (Springer, 2000)

Chudnovski and Chudnovski, Classical and Quantum Models and Arithmetic Problems, Lecture Notes in Pure and Applied Mathematics 92.

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

N. Dattani, Quadratization in discrete optimization and quantum mechanics (open-source, 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."

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