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..."]

"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)."

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."

V.L.Cartas, "The Riemann zeta function applied to the glassy systems and neural networks" (presented at International Conference on Theoretical Physics - Paris, UNESCO, 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."

[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."

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, 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."

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. Ottarsson, "The Ladder Hypothesis"

[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 p_l 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 p_l 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."

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."

K.K. Nambiar, "Electrical equivalent of Riemann Hypothesis"

[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."

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."

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, "Series representations of the Riemann and Hurwitz zeta functions and series and integral representations of the first Stieltjes constant" (preprint 06/2011)

"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."

A. Kuznetsov, "Integral representations for the Dirichlet L-functions and their expansions in Meixner-Pollaczek polynomials and Pochhammer functions" (preprint 2006)

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

C. Nash and D. O'Connor, "Determinants of Laplacians, the Ray-Singer torsion on lens spaces and the Riemann zeta function"

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.

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."

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."

H. Frisk and S. de Gosson, "On the motion of zeros of zeta functions"

"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."

I. Mikoss, "The prime numbers hidden symmetric structure and its relation to the twin prime infinitude and an improved prime number theorem (preprint 2006)

[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."

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."

M. Mulase, "Lectures on the asymptotic expansion of a hermitian matrix integral"

"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."

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

Frantisek Slanina's page on "Number theory & informatics & codes"

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

Wikipedia article on Langlands Program

Wikipedia article on Robert Langlands

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