curiosities of arithmetic gases statistical mechanical systems with an exponential density of states are considered the arithmetic analog of parafermions of arbitrary order is constructed and a formula for boson parafermion equivalence is obtained using properties of the riemann zeta function interactions nontrivial mixing among arithmetic gases using the concept of twisted convolutions are also introduced examples of exactly solvable models are discussed in detail

on the zeros of the ramanujan t dirichlet series in the critical strip we describe computations which show that each of the first 12069 zeros of the ramanujan r dirichlet series of the form it in the region 0 t 6397 is simple and lies on the line 6 the failures of gram s law in this region are also noted the first 5018 zeros and 2228 successive zeros beginning with the 20001st zero are also calculated the distribution of the normalized spacing of the zeros is examined and it appears to be that of the eigenvalues of random matrices these comptuations are done with a berry keating formula for the r dirichlet series and evaluated using mathematica

on the zeros of the riemann zeta function in the critical strip iii we describe extensive computations which show that riemann s zeta function t s has exactly 300 000 001 zeros of the form a it in the region 0 t 119 590 809 282 all these zeros are simple and lie on the line a this extends a similar result for the first 200 000 001 zeros established by brent van de lune te riele and winter in math comp v 39 1982 pp 681 688 counts of the numbers of gram blocks of various types and the failures of rosser s rule are given together with some graphs of the function z t near the first observed failures of rosser s rule

primes in short intervals contrary to what would be predicted on the basis of cramer s model con cerning the distribution of prime numbers we develop evidence that the distribution of x h x for 0 x n is approximately normal with mean h and variance h log n h when n h n 1

random matrix theory and the zeros of ζ s we study the density of the roots of the derivative of the characteristic polynomial z u z of an n n random unitary matrix with distribution given by haar measure on the unitary group based on previous random matrix theory models of the riemann zeta function s this is expected to be an accurate description for the horizontal distribution of the zeros of s to the right of the critical line we show that as n the fraction of the roots of z u z that lie in the region 1 x n 1 z 1 tends to a limit function we derive asymptotic expressions for this function in the limits x and x 0 and compare them with numerical experiments

patterns in eigenvalues the 70th josiah willard gibbs lecture typical large unitary matrices show remarkable patterns in their eigenvalue distribution these same patterns appear in telephone encryption the zeros of riemann s zeta function a variety of physics problems and in the study of toeplitz operators this paper surveys these applications and what is currently known about the patterns

on the zeros of the riemann zeta function in the critical strip ii we describe extensive computations which show that riemann s zeta function t s has exactly 200 000 001 zeros of the form a it in the region 0 t 81 702 130 19 all these zeros are simple and lie on the line a this extends a similar result for the first 81 000 001 zeros established by brent in math comp v 33 1979 pp 1361 1372 counts of the numbers of gram blocks of various types and the failures of rosser s rule are given

probability laws related to the jacobi theta and riemann zeta functions and brownian excursions this paper reviews known results which connect riemann s integral representations of his zeta function involving jacobi s theta function and its derivatives to some particular probability laws governing sums of independent exponential variables these laws are related to one dimensional brownian motion and to higher dimensional bessel processes we present some characterizations of these probability laws and some approximations of riemann s zeta function which are related to these laws

chebyshev centers and some geometric properties of banach spaces abstract in this paper we study two properties p 1 and p 2 introduced in the literature for studying the existence and stability of relative chebyshev centers we reformulate these properties in a way that leads naturally to our results we mainly relate these properties with some geometric properties of banach spaces in particular reflexive spaces having the kadec klee property are characterized in terms of property p 1 and uniform convexity is characterized in terms of property p 2 some continuity results for the center map are also presented

level spacings distribution for large random matrices gaussian fluctuations we study the level spacings distribution for eigenvalues of large n x n matrices from the classical compact groups in the scaling limit when the mean distance between nearest eigenvalues equals 1 defining by 77n s the number of nearest neighbors spacings greater than s 0 smaller than s 0 we prove functional limit theorem for the process 77n s e n s n1 2 giving weak convergence of this distribution to some gaussian random process on 0 x the limiting gaussian random process is universal for all classical compact groups it is holder continuous with any exponent less than 1 2 similar results can be obtained for the n level spacings distribution

random matrix ensembles associated to compact symmetric spaces we introduce random matrix ensembles that correspond to the infinite families of irreducible riemannian symmetric spaces of type i in particular we recover the circular orthogonal and symplectic ensembles of dyson and find other families of unitary orthogonal and symplectic ensembles of jacobi type we discuss the universal and weakly universal features of the global and local correlations of the levels in the bulk and at the hard edge of the spectrum i e at the central points 1 on the unit circle previously known results are extended and we find new simple formulas for the bessel kernels that describe the local correlations at a hard edge

painleve equations nonlinear special functions the six painleve equations pi pvi were first discovered about a hundred years ago by painleve and his colleagues in an investigation of nonlinear second order ordinary differential equations recently there has been considerable interest in the painleve equations primarily due to the fact that they arise as reductions of the soliton equations which are solvable by inverse scattering consequently the painleve equations can be regarded as completely integrable equations and possess solutions which can be expressed in terms of solutions of linear integral equations despite being nonlinear equations although first discovered from strictly mathematical considerations the painleve equations have arisen in a variety of important physical applications including statistical mechanics plasma physics nonlinear waves quantum gravity quantum field theory general relativity nonlinear optics and fibre optics the painleve equations may be thought of a nonlinear analogues of the classical special functions they possess hierarchies of rational solutions and one parameter families of solutions expressible in terms of the classical special functions for special values of the parameters further the painleve equations admit symmetries under affine weyl groups which are related to the associated backlund transformations in this paper i discuss some of the remarkable properties which the painleve equations possess including connection formulae backlund transformations associated discrete equations and hierarchies of exact solutions in particular the second painleve equation pii is used to illustrate these properties and some of the applications of pii are also discussed

applications of the l functions ratios conjectures in upcoming papers by conrey farmer and zirnbauer there appear conjectural formulas for averages over a family of ratios of products of shifted l functions in this paper we will present various applications of these ratios conjectures to a wide variety of problems that are of interest in number theory such as lower order terms in the zero statistics of l functions mollified moments of l functions and discrete averages over zeros of the riemann zeta function in particular using the ratios conjectures we easily derive the answers to a number of notoriously difficult computations

positivity for equations involving polyharmonic operators with dirichlet boundary conditions cranston fabes and zhao 26 5 established the uniform bound sup x y 2 x 6 y r g1 n x z g1 n z y dz g1 n x y m 1 1 where g1 n x y is the green function for the laplacian with dirichlet boundary conditions on a lipschitz domain rn with n 3 see 27 for n 2 this estimate was used in 23 and 18 to obtain positivity uniformly with respect to f 0 for noncooperative elliptic systems as

extremal functions in de branges and euclidean spaces ii this paper presents the gaussian subordination framework to generate optimal one sided approximations to multidimensional real valued functions by functions of prescribed exponential type such extremal problems date back to the works of beurling and selberg and provide a variety of applications in analysis and analytic number theory here we majorize and minorize on bbb r n the gaussian bf x mapsto e pi lambda bf x 2 where lambda0 is a free parameter by functions with distributional fourier transforms supported on euclidean balls optimizing weighted l 1 errors by integrating the parameter lambda against suitable measures we solve the analogous problem for a wide class of radial functions applications to inequalities and periodic analogues are discussed the constructions presented here rely on the theory of de branges spaces of entire functions and on new interpolation tools derived from the theory of laplace transforms of laguerre p o lya functions

low lying zeros of l functions and random matrix theory by looking at the average behavior n level density of the low lying zeros of certain families of l functions we find evidence as predicted by function field analogs in favor of a spectral interpretation of the nontrivial zeros in terms of the classical compact groups

self organized criticality attributed to a central limit like convergence effect abstract self organized criticality is a hypothesis used to explain the origin of 1 f noise and other scaling behaviors despite being proposed nearly 30 years ago no consensus exists as to its exact definition or mathematical mechanism s recently a model for 1 f noise was proposed based on a family of statistical distributions known as the tweedie exponential dispersion models these distributions are characterized by an inherent scale invariance that manifests as a variance to mean power law called fluctuation scaling they also serve as foci of convergence in a limit theorem on independent and identically distributed distributions fluctuation scaling can be modeled by self similar stochastic processes that relate the variance to mean power law to 1 f noise through their correlation structure a hypothesis is proposed whereby the effects of self organized criticality are mathematically modeled by the tweedie distributions and their convergence behavior as applied to self similar stochastic processes sandpile model fluctuations are shown to manifest 1 f noise fluctuation scaling and to conform to the tweedie compound poisson distribution the tweedie models and their convergence theorem allow for a mechanistic explanation of 1 f noise and fluctuation scaling in phenomena conventionally attributed to self organized criticality thus providing a paradigm shift in our understanding of these phenomena

an improved lower bound for the de bruijn newman constant in this article we report on computations that led to the discovery of a new lehmer pair of zeros for the riemann function given this new close pair of zeros we improve the known lower bound for de bruijn newman constant a the riemann hypothesis is equivalent to the assertion a 1 14541 x 10 11 this new bound confirms the belief that if the riemann hypothesis is true it is barely true

random matrices generalized zeta functions and self similarity of zero distributions there is growing evidence for a connection between random matrix theories used in physics and the theory of the riemann zeta function and l functions the theory underlying the location of the zeros of these generalized zeta functions is one of the key unsolved problems physicists are interested because of the hilbert polya conjecture that the non trivial zeros of the zeta function correspond to the eigenvalues of some positive operator to complement the continuing theoretical work it would be useful to study empirically the locations of the zeros by different methods in this paper we use the rescaled range analysis to study the spacings between successive zeros of these functions over large ranges of the zeros the spacings have a hurst exponent of about 0 095 using sample sizes of 10 000 zeros this implies that the distribution has a high fractal dimension 1 9 and shows a lot of detailed structure the distribution is of the anti persistent fractional brownian motion type with a significant degree of anti persistence thus the high order zeros of these functions show a remarkable self similarity in their distribution over fifteen orders of magnitude for the riemann zeta function we find that the hurst exponents for the random matrix theories show a different behaviour a heuristic study of the effect of low order primes seems to show that this effect is a promising candidate to explain the results that we observe in this study we study the distribution of zeros for l functions of conductors 3 and 4 and find that the distribution is similar to that of the riemann zeta functions

interpolation formulas with derivatives in de branges spaces ii abstract we investigate necessary and sufficient conditions under which entire functions in de branges spaces can be recovered from function values and values of derivatives our main focus is on spaces with a structure function whose logarithmic derivative is bounded in the upper half plane

on the pair correlation of zeros of the riemann zeta function in 1972 montgomery 20 21 introduced a new method for studying the zeros of the riemann zeta function one of his main accomplishments was to determine partially the pair correlation of zeros and to apply his results to obtain new information on multiplicity of zeros and gaps between zeros perhaps more importantly he conjectured on number theoretic grounds an asymptotic formula for the pair correlation of zeros and found that the form of this correlation exactly agreed with the gaussian unitary ensemble gue model for random hermitian matrices which had been studied earlier by physicists he was therefore able to formulate a general n correlation conjecture for zeros during the 1980s odlyzko 23 24 performed extensive numerical calculations of the correlations for zeros in ranges up to the 10th zero and found excellent agreement with the gue model more recently hejhal 17 was able to prove the same partial result for triple correlation as montgomery proved for pair correlation and rudnick and sarnak 25 have done the same for n correlation rudnick and sarnak extended their results to a large class of l functions and also showed that the riemann hypothesis rh is not needed for smoother forms of the asymptotic result very recently bogomolny and keating 1 2 have used a prime twin type conjecture to derive heuristically the n correlation conjecture beyond the range where the results of rudnick and sarnak apply the conclusion of all this work is to rmly establish but not prove the gue distribution for zeros of many zeta functions there is a dual relationship between zeros of the riemann zeta function and prime numbers following montgomery s work it was realized that information on pair correlation of zeros could be used to obtain information on primes this connection was developed by gallagher and mueller 10 and heath brown 15 later goldston and montgomery 14 found an equivalence under the riemann hypothesis between the pair correlation of zeros and the variance for the number of primes in short intervals this equivalence arises out of the explicit formula via a parseval relation together with a tauberian theorem from this work one sees that to obtain new information on primes from zeros will require some new insight on the zeros and while the connections to statistical physics mentioned above are a possible source of this insight so far no progress has been made on this fundamental problem the motivation for this paper is the observation that there are additional tools

classical solutions for some higher order semilinear elliptic equations under weak growth conditions we want to explain the crucial difference between the second order m 1 and higher order m 1 case if m 1 a very satisfactory result is known indeed the sign condition 2 alone is sufficient to ensure classical solvability of the dirichlet problem 1 there are two very strong devices in the theory of second order elliptic equations which make the proof of the needed a priori maximum estimate an easy exercise

the bakerian lecture 1987 quantum chaology bounded or driven classical systems often exhibit chaos exponential instability that persists but their quantum counterparts do not nevertheless there are new regimes of quantum behaviour that emerge in the semiclassical limit and depend on whether the classical orbits are regular or chaotic and this motivates the following definition definition quantum chaology is the study of semiclassical but nonclassical behaviour characteristic of systems whose classical motion exhibits chaos this is illustrated by the statistics of energy levels on scales comparable with the mean level spacing of order h n for n freedoms these fall into universality classes for classically chaotic systems the statistics are those of random matrices real symmetric or complex hermitian depending on the presence or absence of time reversal symmetry for classically regular ones the statistics are poisson on larger scales of order h i e classically small but semiclassically large universality breaks down these phenomena are being explained by representing spectra in terms of classical closed orbits universal spectral behaviour has its origin in very long orbits non universal behaviour depends only on short ones

extreme gaps between eigenvalues of random matrices this paper studies the extreme gaps between eigenvalues of random matrices we give the joint limiting law of the smallest gaps for haar distributed unitary matrices and matrices from the gaussian unitary ensemble in particular the kth smallest gap normalized by a factor n 4 3 has a limiting density proportional to x3k 1e x3 concerning the largest gaps normalized by n logn they converge in lp to a constant for all p 0 these results are compared with the extreme gaps between zeros of the riemann zeta function

asymptotics of the largest zeros of some orthogonal polynomials we study the asymptotics of the largest zeros of the wilson hermite and q laguerre polynomials using two distinct techniques the first is based on the coulomb fluid technique developed in a previous paper where the primary input is the weight function while the second uses the method of chain sequences which supplies inequalities for the largest zeros using the recurrence coefficients we also investigate the asymptotics of the largest zeros of the polynomials orthogonal to the weight functions for c 0 and m a positive even integer

finite size corrections in random matrix theory and odlyzko s dataset for the riemann zeros odlyzko has computed a dataset listing more than 109 successive riemann zeros starting from a zero number to beyond 1023 this dataset relates to random matrix theory as according to the montgomery odlyzko law the statistical properties of the large riemann zeros agree with the statistical properties of the eigenvalues of large random hermitian matrices moreover keating and snaith and then bogomolny and co workers have used n n random unitary matrices to analyse deviations from this law we contribute to this line of study in two ways first we point out that a natural process to apply to the dataset is to minimize it by deleting each member independently with some specified probability and we proceed to compute empirical two point correlation functions and nearest neighbour spacings in this setting second we show how to characterize the order 1 n 2 correction term to the spacing distribution for random unitary matrices in terms of a second order differential equation with coefficients that are painleve transcendents and where the thinning parameter appears only in the boundary condition this equation can be solved numerically using a power series method in comparison to the riemann zero data accurate agreement is exhibited

on the zeros of the riemann zeta function in the critical strip iv etude de la localisation des zeros complexes de la fonction zeta de riemann verification numerique de l hypothese de riemann pour les 1 500 000 001 premiers zeros de la forme it dans la bande critique avec 0 t 545 439 823 215 tous ces zeros ayant pour partie reelle 1 2

a sparser matrix representation of the mertens function abstract redheffer introduced a sparse 0 1 matrix whose determinant provides an unorthodox representation of the mertens function the growth of the mertens function is intimately tied to the locations of the nontrivial zeros of the riemann zeta function we introduce a 0 1 matrix whose determinant also provides a nontrivial representation of the mertens function but it is sparser than the matrix of redheffer it is sparser in two ways the matrix has fewer nonzero entries o n compared to o n log n and it has substantially fewer eigenvalues different from 1 this construction is a corollary of a more general result that holds for multiplicative sequences

on the singular values of gaussian random matrices this short note is about the singular value distribution of gaussian random matrices i e gaussian ensemble or ge of size n we present a new approach for deriving the p d f of the singular values directly from the singular value decomposition svd form which also takes advantage of the rotational invariance of ge and the lie algebra of the orthogonal group our method is direct and more general than the conventional approach that relies on the wishart ensemble and the combination of qr and cholesky decomposition directly based on this p d f and its interpretation by statistical mechanics we give the physics proof that in the thermodynamic limit n the singular value distribution satisfies the quadrant law similar to the celebrated semi circle lawestablished by wigner more than 40 years ago for the spectral distribution of gaussian orthogonal or unitary ensembles this quadrant law was also proved earlier and mathematically more rigorously by some authors based on probabilistic estimations and the moment method but not directly from the p d f formula 2001 elsevier science inc all rights reserved

existence of solutions to nonlinear subcritical higher order elliptic dirichlet problems abstract we consider the 2 m th order elliptic boundary value problem l u f x u on a bounded smooth domain r n with dirichlet boundary conditions on the operator l is a uniformly elliptic linear operator of order 2 m whose principle part is of the form i j 1 n a i j x 2 x i x j m we assume that f is superlinear at the origin and satisfies lim s f x s s q h x lim s f x s s q k x where h k c are positive functions and q 1 is subcritical by combining degree theory with new and recently established a priori estimates we prove the existence of a nontrivial solution

gaps between zeros of ζ s and the distribution of zeros of ζ s abstract we assume the riemann hypothesis in this paper we settle a conjecture of farmer and ki in a stronger form roughly speaking we show that there is a positive proportion of small gaps between consecutive zeros of the zeta function s if and only if there is a positive proportion of zeros of s lying very closely to the half line our work has applications to the siegel zero problem we provide a criterion for the non existence of the siegel zero solely in terms of the distribution of the zeros of s finally on the riemann hypothesis and the pair correlation conjecture we obtain near optimal bounds for the number of zeros of s lying very closely to the half line such bounds are relevant to a deeper understanding of levinson s method allowing us to place one third of the zeros of the riemann zeta function on the half line

fast algorithms for multiple evaluations of the riemann zeta function the best previously know algorithm for evaluating the riemann zeta function it with bounded and t large to moderate accuracy was based on the rienmann siegel formula and required on the order of t 1 2 operations for each value that was computed new algorithms are presented in this paper which enable one to compute any single value of it with fixed and t t t t 1 2 in o te operations on numbers of o log t bits for any e 0 for example these algorithms lead to methods for numerically verifying the riemann hypothesis for the first n zeros in what is expected to be o n 1 e operations

duality biorthogonal polynomials and multi matrix models the statistical distribution of eigenvalues of pairs of coupled random matrices can be expressed in terms of integral kernels having a generalized christoffel darboux form constructed from sequences of biorthogonal polynomials for measures involving exponentials of a pair of polynomials v1 v2 in two different variables these kernels may be expressed in terms of finite dimensional windows spanned by finite subsequences having length equal to the degree of one or the other of the polynomials v1 v2 the vectors formed by such subsequences satisfy dual pairs of first order systems of linear differential equations with polynomial coefficients having rank equal to one of the degrees of v1 or v2 and degree equal to the other they also satisfy recursion relations connecting the consecutive windows and deformation equations determining how they change under variations in the coefficients of the polynomials v1 and v2 viewed as overdetermined systems of linear difference differential deformation equations these are shown to be compatible and hence to admit simultaneous fundamental systems of solutions the main result is the demonstration of a spectral duality property namely that the spectral curves defined by the characteristic equations of the pair of matrices defining the dual differential systems are equal upon interchange of eigenvalue and polynomial parameters

on the distribution of the total energy of a system of non interacting fermions random matrix and semiclassical estimates abstract we consider a single particle spectrum as given by the eigenvalues of the wigner dyson ensembles of random matrices and fill consecutive single particle levels with n fermions assuming that the fermions are non interacting we show that the distribution of the total energy is gaussian and its variance grows as n 2 log n in the large n limit next to leading order corrections are also computed some related quantities are discussed in particular the nearest neighbor spacing autocorrelation function canonical and grand canonical approaches are considered and compared in detail a semiclassical formula describing as a function of n a non universal behavior of the variance of the total energy starting at a critical number of particles is also obtained it is illustrated with the particular case of single particle energies given by the imaginary part of the zeros of the riemann zeta function on the critical line

computational strategies for the riemann zeta function we provide a compendium of evaluation methods for the riemann zeta function presenting formulae ranging from historical attempts to recently found convergent series to curious oddities old and new we concentrate primarily on practical computational issues such issues depending on the domain of the argument the desired speed of computation and the incidence of what we call value recycling c 2000 elsevier science b v all rights reserved

matrix models for beta ensembles this paper constructs tridiagonal random matrix models for general 0 hermite gaussian and laguerre wishart ensembles these generalize the well known gaussian and wishart models for 1 2 4 furthermore in the cases of the laguerre ensembles we eliminate the exponent quantization present in the previously known models we further discuss applications for the new matrix models and present some open problems

explicit bounds for primality testing and related problems many number theoretic algorithms rely on a result of ankeny which states that if the extended riemann hypothesis erh is true any nontrivial multiplicative subgroup of the integers modulo m omits a number that is o log2 m this has been generalized by lagarias montgomery and odlyzko to give a similar bound for the least prime ideal that does not split completely in an abelian extension of number fields this paper gives a different proof of this theorem in which explicit constants are supplied the bounds imply that if the erh holds a composite number m has a witness for its compositeness in the sense of miller or solovay strassen that is at most 2 log2m

approaching word length distribution via level spectra abstract treating a text after the removal of paragraphs and punctuations as a spectrum of blanks the distributions of the length of words of ten languages are analyzed using models from the statistical theory of spectra it is found that the ten languages can be classified into two families one with words that follow a wigner like distribution while the words of the other obey a poisson like distribution

linear law for the logarithms of the riemann periods at simple critical zeta zeros each simple zero 1 2 i n of the riemann zeta function on the critical line with n 0 is a center for the flow s s of the riemann xi function with an associated period t n it is shown that as n numerical evaluation leads to the conjecture that this inequality can be replaced by an equality assuming the riemann hypothesis and a zeta zero separation conjecture n for some exponent 0 we obtain the upper bound log t n assuming a weakened form of a conjecture of gonek giving a bound for the reciprocal of the derivative of zeta at each zero we obtain the expected upper bound for the periods so conditionally log t n 4 n o log n indeed this linear relationship is equivalent to the given weakened conjecture which implies the zero separation conjecture provided the exponent is sufficiently large the frequencies corresponding to the periods relate to natural eigenvalues for the hilbert polya conjecture they may provide a goal for those seeking a self adjoint operator related to the riemann hypothesis

derivative moments for characteristic polynomials from the cue we calculate joint moments of the characteristic polynomial of a random unitary matrix from the circular unitary ensemble and its derivative in the case that the power in the moments is an odd positive integer the calculations are carried out for finite matrix size and in the limit as the size of the matrices goes to infinity the latter asymptotic calculation allows us to prove a long standing conjecture from random matrix theory

hilbert spaces and the pair correlation of zeros of the riemann zeta function montgomery s pair correlation conjecture predicts the asymptotic behavior of the function n t defined to be the number of pairs and of ordinates of nontrivial zeros of the riemann zetafunction satisfying 0 0 using montgomery s formula and some extremal functions of exponential type these functions are optimal in the sense that they majorize and minorize the characteristic function of the interval in a way to minimize the l1 r 1 sin x x 2 dx error we give a complete solution for this extremal problem using the framework of reproducing kernel hilbert spaces of entire functions this extends previous work of p x gallagher 18 in 1985 where the case 1 2 n was considered using non extremal majorants and minorants

on the histogram as a density estimator l 2 theory let f be a probability density on an interval i finite or infinite i includes its finite endpoints if any and f vanishes outside of i let x1 x k be independent random variables with common density f the empirical histogram for the x s is often used to estimate f to define this object choose a reference point xosi and a cell width h let nj be the number of x s falling in the j th class interval

on the distribution of imaginary parts of zeros of the riemann zeta function ii mathematics subject classification 2000 primary 11m26 secondary 11k38 we continue our investigation of the distribution of the fractional parts of whereis a fixed non zero real number and runs over the imaginary parts of the non trivial zeros of the riemann zeta function we establish some connections to mont gomery s pair correlation function and the distribution of primes in short intervals we also discuss analogous results for a more general l function

pair correlation estimates for the zeros of the zeta function via semidefinite programming abstract in this paper we study the distribution of the non trivial zeros of the riemann zeta function s and other l functions using montgomery s pair correlation approach we use semidefinite programming to improve upon numerous asymptotic bounds in the theory of s including the proportion of distinct zeros counts of small gaps between zeros and sums involving multiplicities of zeros

extremal functions in de branges and euclidean spaces abstract in this work we obtain optimal majorants and minorants of exponential type for a wide class of radial functions on r n these extremal functions minimize the l 1 r n x 2 2 n d x distance to the original function where 1 is a free parameter to achieve this result we develop new interpolation tools to solve an associated extremal problem for the exponential function f x e x where 0 in the general framework of de branges spaces of entire functions we then specialize the construction to a particular family of homogeneous de branges spaces to approach the multidimensional euclidean case finally we extend the result from the exponential function to a class of subordinated radial functions via integration on the parameter 0 against suitable measures applications of the results presented here include multidimensional versions of hilbert type inequalities extremal one sided approximations by trigonometric polynomials for a class of even periodic functions and extremal one sided approximations by polynomials for a class of functions on the sphere s n 1 with an axis of symmetry

on the distribution of spacings between zeros of the zeta function etude numerique de la distribution des espacements des zeros de la fonction zeta de riemann observation de certains phenomenes inatendus

disproof of the mertens conjecture the mertens conjecture states that m x 1 where m x n x n and n is the mo bius function this conjecture has attracted a substantial amount of interest in its almost 100 years of existence because its truth was known to imply the truth of the riemann hypothesis this paper disproves the mertens conjecture by showing that x lim sup m x x 2 1 06 the disproof relies on extensive computations with the zeros of the zeta function and does not provide an explicit counterexample disproof of the mertens conjecture a m odlyzko at t bell laboratories murray hill new jersey 07974 usa and h j j te riele centre for mathematics and computer science kruislaan 413 1098 sj amsterdam the netherlands

optimization problems for an elastic plate this paper concerns optimization problems related to bi harmonic equations subject to either navier or dirichlet homogeneous boundary conditions physically in dimension two our equation models the deformation of an elastic plate which is either hinged or clamped along the boundary under load we discuss existence uniqueness and properties of the optimizers

extremal problems in de branges spaces the case of truncated and odd functions in this paper we find extremal one sided approximations of exponential type for a class of truncated and odd functions with a certain exponential subordination these approximations optimize the l 1 mathbb r e x 2 mathrm d x error where e is an arbitrary hermite biehler entire function of bounded type in the upper half plane this extends the work of holt and vaaler duke math j 83 203 247 1996 for the signum function we also provide periodic analogues of these results finding optimal one sided approximations by trigonometric polynomials of a given degree to a class of periodic functions with exponential subordination these extremal trigonometric polynomials optimize the l 1 mathbb r mathbb z mathrm d vartheta error where vartheta is an arbitrary nontrivial measure on mathbb r mathbb z the periodic results extend the work of li and vaaler indiana univ math j 48 1 183 236 1999 who considered this problem for the sawtooth function with respect to jacobi measures our techniques are based on the theory of reproducing kernel hilbert spaces of entire functions and of polynomials and on the construction of suitable interpolations with nodes at the zeros of laguerre polya functions

the riemann zeros and eigenvalue asymptotics comparison between formulae for the counting functions of the heights tn of the riemann zeros and of semiclassical quantum eigenvalues en suggests that the tn are eigenvalues of an unknown hermitean operator h obtained by quantizing a classical dynamical system with hamiltonian hcl many features of hcl are provided by the analogy for example the riemann dynamics should be chaotic and have periodic orbits whose periods are multiples of logarithms of prime numbers statistics of the tn have a similar structure to those of the semiclassical en in particular they display random matrix universality at short range and nonuniversal behaviour over longer ranges very refined features of the statistics of the tn can be computed accurately from formulae with quantum analogues the riemann siegel formula for the zeta function is described in detail its interpretation as a relation between long and short periodic orbits gives further insights into the quantum spectral fluctuations we speculate that the riemann dynamics is related to the trajectories generated by the classical hamiltonian hcl xp

semiclassical theory of spectral rigidity the spectral rigidity l of a set of quantal energy levels is the mean square deviation of the spectral staircase from the straight line that best fits it over a range of l mean level spacings in the semiclassical limit 0 formulae are obtained giving l as a sum over classical periodic orbits when l l max where l max n 1 for a system of n freedoms l is shown to display the following universal behaviour as a result of properties of very long classical orbits if the system is classically integrable all periodic orbits filling tori l 1 5 l as in an uncorrelated poisson eigenvalue sequence if the system is classically chaotic all periodic orbits isolated and unstable and has no symmetry l in l 2 2 d if 1 l l max as in the gaussian unitary ensemble of random matrix theory if the system is chaotic and has time reversal symmetry l in l 2 e if 1 l l max as in the gaussian orthogonal ensemble when l l max l saturates non universally at a value determined by short classical orbits of order n 1 for integrable systems and in 1 for chaotic systems these results are obtained by using the periodic orbit expansion for the spectral density together with classical sum rules for the intensities of long orbits and a semiclassical sum rule restricting the manner in which their contributions interfere for two examples l is studied in detail the rectangular billiard integrable and the riemann zeta function assuming its zeros to be the eigenvalues of an unknown quantum system whose unknown classical limit is chaotic

energy level statistics of the hadamard gutzwiller ensemble abstract the statistical properties of the quantal energy levels of the hadamard gutzwiller ensemble whose classical members belong to the class of systems with hard chaos are investigated based on a sample of 4500 energy levels it is shown that the short range statistics as nearest neighbour spacing distributions are governed by the goe predictions of random matrix theory which was first surmised by wigner and by landau and smorodinsky for nuclear level statistics this result strengthens the hypothesis that quantum systems with chaotic classical counterpart display level repulsion as predicted by random matrix theory however the level statistics describing correlations over greater level distances deviate from the goe predictions which is explained as a simple consequrnce of the fact that the spectral rigidity 3 l introduced by dyson and mehta saturates non universally at a finite value for l in complete agreement with the semiclassical theory developed by berry

self consistent method for density estimation the estimation of a density profile from experimental data points is a challenging problem usually tackled by plotting a histogram prior assumptions on the nature of the density from its smoothness to the specification of its form allow the design of more accurate estimation procedures such as maximum likelihood our aim is to construct a procedure that makes no explicit assumptions but still providing an accurate estimate of the density we introduce the self consistent estimate the power spectrum of a candidate density is given and an estimation procedure is constructed on the assumption to be released a posteriori that the candidate is correct the self consistent estimate is defined as a prior candidate density that precisely reproduces itself our main result is to derive the exact expression of the self consistent estimate for any given dataset and to study its properties applications of the method require neither priors on the form of the density nor the subjective choice of parameters a cutoff frequency akin to a bin size or a kernel bandwidth emerges naturally from the derivation we apply the self consistent estimate to artificial data generated from various distributions and show that it reaches the theoretical limit for the scaling of the square error with the dataset size

the zeta function on the critical line numerical evidence for moments and random matrix theory models results of extensive computations of moments of the riemann zeta function on the critical line are presented calculated values are compared with predictions motivated by random matrix theory the results can help in deciding between those and competing predictions it is shown that for high moments and at large heights the variability of moment values over adjacent intervals is substantial even when those intervals are long as long as a block containing 109 zeros near zero number 1023 more than anything else the variability illustrates the limits of what one can learn about the zeta function from numerical evidence it is shown the rate of decline of extreme values of the moments is modelled relatively well by power laws also some long range correlations in the values of the second moment as well as asymptotic oscillations in the values of the shifted fourth moment are found the computations described here relied on several representations of the zeta function the numerical comparison of their effectiveness that is presented is of independent interest for future large scale computations

time delay resonances riemann zeros and chaos in a model quantum scattering system the quantum treatment of an intrinsically chaotic model scattering system originally studied by gutzwiller 1983 is extended to include the time delay and to make explicit the zeros of the riemann zeta function in the mathematical expressions for the scattering matrix s and the time delay the system consists of a particle moving on a two dimensional surface of constant negative curvature the scattering in this unusual system is dominated by resonances associated with poles of s in the complex momentum plane the real parts of these poles are one half of the imaginary parts of the riemann zeros the resonances have a constant width but their average spacing varies with the momentum the focal point is the manifestation of chaotic behaviour in the time delay features considered in this regard include the momentum dependence of the time delay in regions of overlapping and isolated resonances the decomposition of the time delay into an average dynamical component and a fluctuating chaotic component and characterisation of the fluctuating component by its autocorrelation function

compactly uniformly convex spaces and property β of rolewicz abstract we study different facets of property of rolewicz we remark that the notions of compact uniform convexity and property are isometrically equivalent and present new examples of spaces with that property an observation is made that the property can be formulated in terms of graphs and an estimate of the modulus is also given

lorentz invariant hamiltonian and riemann hypothesis we have given some arguments that a two dimensional lorentz invariant hamiltonian may be relevant to the riemann hypothesis concerning zero points of the riemann zeta function some eigenfunction of the hamiltonian corresponding to infinite dimensional representation of the lorentz group have many interesting properties especially a relationship exists between the zero zeta function condition and the absence of trivial representations in the wavefunction we also give a heuristic argument for the validity of the hypothesis

the p and h p versions of the finite element method basic principles and properties in the classical form of the finite element method called the hversion piecewise polynomials of fixed degree p are used and the mesh size h is decreased for accuracy in this paper we discuss the fundamental theoretical ideas behind the relatively recent p version and h p version in the p version a fixed mesh is used and p is allowed to increase the h p version combines both approaches the authors describe and explain the basic properties and characteristics of these newer versions especially in areas where their behavior is significantly different from that of the h version simplified proofs of key concepts are included and computational illustrations of several results are provided a benchmark comparison between the various versions in included

the effect of convolving families of l functions on the underlying group symmetries let fn and gm be families of primitive automorphic l functions for gln aq and glm aq respectively such that as n m the statistical behavior 1 level density of the low lying zeros of l functions in fn resp gm agrees with that of the eigenvalues near 1 of matrices in g1 resp g2 as the size of the matrices tend to infinity where each gi is one of the classical compact groups unitary u symplectic sp or orthogonal o so even so odd assuming that the convolved families of l functions fn gm are automorphic we study their 1 level density we also study convolved families of the form f gm for a fixed f under natural assumptions on the families which hold in many cases we can associate to each family l of l functions a symmetry constant cl equal to 0 resp 1 or 1 if the corresponding low lying zero statistics agree with those of the unitary resp symplectic or orthogonal group our main result is that cf g cf cg the symmetry type of the convolved family is the product of the symmetry types of the two families a similar statement holds for the convolved families f gm we provide examples built from dirichlet l functions and holomorphic modular forms and their symmetric powers an interesting special case is to convolve two families of elliptic curves with positive rank in this case the symmetry group of the convolution is independent of the ranks in accordance with the general principle of multiplicativity of the symmetry constants but the ranks persist before taking the limit n m as lower order terms

existence of chebyshev centers best n nets and best compact approximants in this paper we investigate the existence and continuity of chebyshev centers best n nets and best compact sets some of our positive results were obtained using the concept of quasi uniform convexity furthermore several examples of nonexistence are given e g a sublattice m of c o 1 and a bounded subset b c m is constructed which has no chebyshev center no best n net and not best compact set approximant

detecting squarefree numbers we present an algorithm based on the explicit formula for l functions and conditional on grh for proving that a given integer is squarefree with little or no knowledge of its factorization we analyze the algorithm both theoretically and practically and use it to prove that several rsa challenge numbers are not squarefull f a non zero element of k x then it is well known that f is squarefree if and only if gcd f f 0 1 and the latter condition may be checked quickly in deterministic polynomial time by the euclidean algorithm it is a long standing question in algorithmic number theory whether there is a correspondingly simple procedure to test if a given integer is squarefree in particular can one determine whether n 2 z is squarefree more rapidly than by factoring it in this paper we describe an algorithm conditional on the generalized riemann hy pothesis grh for proving an integer squarefree with little or no knowledge of its factor ization and analyze the complexity of the algorithm both theoretically and practically in particular we present some heuristic evidence based on random matrix theory and other probabilistic calculations that our algorithm runs in deterministic subexponential time o exp logn 2 3 o 1 although this is poorer than the performance expected of the cur rent best known factoring algorithms our method is able to give partial results that one does not obtain from a failed attempt at factoring in particular we show the following see

on fluctuations of riemann s zeta zeros it is shown that the normalized fluctuations of riemann s zeta zeros around their predicted locations follow the gaussian law it is also shown that fluctuations of two zeros gamma k and gamma k x with x sim left log k right beta beta 0 for large k follow the two variate gaussian distribution with correlation left 1 beta right

on the quantum reconstruction of the riemann zeros we discuss a possible spectral realization of the riemann zeros based on the hamiltonian h xp perturbed by a term that depends on two potentials which are related to the berry keating semiclassical constraints we find perturbatively the potentials whose jost function is given by the zeta function it for 1 for 1 2 we find the potentials that yield the smooth approximation to the zeros we show that the existence of potentials realizing the zeta function at 1 2 as a jost function would imply that the riemann zeros are point like spectra embedded in the continuum resolving in that way the emission spectral interpretation

the riemann siegel expansion for the zeta function high orders and remainders on the critical line s i t t real riemann s zeta function can be calculated with high accuracy by the riemann siegel expansion this is derived here by elementary formal manipulations of the dirichlet series it is shown that the expansion is divergent with the high orders r having the familiar 9factorial9 divided by power9 dependence decorated with an unfamiliar slowly varying multiplier function which is calculated explicitly terms of the series decrease until r r 2 t and then increase the form of the remainder when the expansion is truncated near r is determined it is of order exp t indicating that the critical line is a stokes line for the riemann siegel expansion these conclusions are supported by computations of the first 50 coefficients in the expansion and of the remainders as a function of truncation for several values of t