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

zeroes of zeta functions and symmetry hilbert and polya suggested that there might be a natural spectral interpretation of the zeroes of the riemann zeta function while at the time there was little evidence for this today the evidence is quite convincing firstly there are the function field analogues that is zeta functions of curves over finite fields and their generalizations for these a spectral interpretation for their zeroes exists in terms of eigenvalues of frobenius on cohomology secondly the developments both theoretical and numerical on the local spacing distributions between the high zeroes of the zeta function and its generalizations give striking evidence for such a spectral connection moreover the low lying zeroes of various families of zeta functions follow laws for the eigenvalue distributions of members of the classical groups in this paper we review these developments in order to present the material fluently we do not proceed in chronological order of discovery also in concentrating entirely on the subject matter of the title we are ignoring the standard body of important work that has been done on the zeta function and l functions 1 the montgomery odlyzko law we begin with the riemann zeta function and some phenomenology associated with it

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

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

on some techniques for approximating boundary conditions in the finite element method we discuss the stabilization of finite element methods in which essential boundary conditions are approximated by babuska s method of lagrange multipliers and we show that there is a close connection with this technique and a classical method by nitsche

improved multiquadric method for elliptic partial differential equations via pde collocation on the boundary the multiquadric radial basis function mq method is a recent meshless collocation method with global basis functions it was introduced for discretizing partial differential equations pdes by kansa in the early 1990s the mq method was originally used for interpolation of scattered data and it was shown to have exponential convergence for interpolation problems in 1 we have extended the kansa mq method to numerical solution and detection of bifurcations in 1d and 2d parameterized nonlinear elliptic pdes we have found there that the modest size nonlinear systems resulting from the mq discretization can be efficiently continued by a standard continuation software such as auto we have observed high accuracy with a small number of unknowns as compared with most known results from the literature in this paper we formulate an improved kansa mq method with pde collocation on the boundary mq pdecb we add an additional set of nodes which can lie inside or outside of the domain adjacent to the boundary and correspondingly add an additional set of collocation equations obtained via collocation of the pde on the boundary numerical results are given that show a considerable improvement in accuracy of the mq pdecb method over the kansa mq method with both methods having exponential convergence with essentially the same rates

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

a meshless galerkin method for dirichlet problems using radial basis functions in this paper a numerical method is given for partial differential equations which combines the use of lagrange multipliers with radial basis functions it is a new method to deal with difficulties that arise in the galerkin radial basis function approximation applied to dirichlet also mixed boundary value problems convergence analysis results are given several examples show the efficiency of the method using tps or sobolev splines

ramanujan hardy littlewood riesz phenomena for hecke forms abstract we generalize a result of ramanujan hardy and littlewood to the setting of primitive hecke forms which interestingly exhibits faster convergence than in the initial case of the riemann zeta function we also provide a criterion in the spirit of riesz for the riemann hypothesis for the associated l functions

on unsymmetric collocation by radial basis functions solving partial differential equations by collocation with radial basis functions can be efficiently done by a technique first proposed by kansa in 1990 it rewrites the problem as a generalized interpolation problem and the solution is obtained by solving a possibly large linear system the method has been used successfully in a variety of applications but a proof of nonsingularity of the linear system was still missing this paper shows that a general proof of this fact is impossible however numerical evidence shows that cases of singularity are rare and have to be constructed with quite some effort

boundary lagrange multipliers in finite element methods error analysis in natural norms a finite element procedure for circumventing the babuska brezzi condition in mixed formulations with lagrange multipliers defined on the boundary is presented residual terms constructed from the euler lagrange equations are added to the classical galerkin formulation in order to attain coercivity in a mesh dependent norm convergence is proven for the primal variable and the multiplier in the natural mesh independent norm of the problem generalizing results of a previous paper

on the zeros of zeta s near the critical line let rho prime beta prime i gamma prime denote the zeros of zeta prime s s sigma it it is shown that there is a positive proportion of the zeros of zeta prime s in 0 t t satisfying beta prime 1 2 ll log t 1 further results relying on the riemann hypothesis and conjectures on the gaps between the zeros of zeta s are also obtained

on mean values of mollifiers and l functions associated to primitive cusp forms we study the second moment of the l function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers this family is a natural extension of the mollifers considered by conrey and by bui conrey and young as an application we improve the current lower bound on critical zeros of holomorphic primitive cusp forms

a note on the meshless method using radial basis functions in this paper we prove some inverse inequalities for the finite dimensional subspace interpolated by radial basis functions the properties of these inequalities are worse than those in finite element spaces these bad properties result in a huge condition number for both the collocation method and galerkin method using radial basis functions as applications theoretical estimates of the condition numbers of a meshless method using radial basis functions are given

zeros of normalized combinations of ξ k s on re s 1 2 abstract we consider functions of the form f c a t s j 0 m c j 1 j l 2 j a 2 j s with l log t 2 and c j real constants satisfying a certain constraint we show that as t the proportion of zeros of f c a t s on the critical line re s 1 2 tends to 1

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

forced oscillations of a massive point on a compact surface with a boundary abstract we present sufficient conditions for the existence of a periodic solution for a class of systems describing the periodically forced motion of a massive point on a compact surface with a boundary

a particle partition of unity method for the solution of elliptic parabolic and hyperbolic pdes in this paper we present a meshless discretization technique for instationary convection diffusion problems it is based on operator splitting the method of characteristics and a generalized partition of unity method we focus on the discretization process and its quality the method may be used as an h version or a p version even for general particle distributions the convergence behavior of the different versions corresponds to that of the respective version of the finite element method on a uniform grid we discuss the implementational aspects of the proposed method furthermore we present the results of numerical examples where we considered instationary convection diffusion instationary diffusion linear advection and elliptic problems

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

meshless galerkin methods using radial basis functions we combine the theory of radial basis functions with the field of galerkin methods to solve partial differential equations after a general description of the method we show convergence and derive error estimates for smooth problems in arbitrary dimensions

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

the natural radial element method summary in this work an innovative numerical approach is proposed which combines the simplicity of low order finite elements connectivity with the geometric flexibility of meshless methods the natural neighbour concept is applied to enforce the nodal connectivity resorting to the delaunay triangulation a background integration mesh is constructed completely dependent on the nodal mesh the nodal connectivity is imposed through nodal sets with reduce size reducing significantly the test function construction cost the interpolations functions constructed using euclidian norms are easily obtained to prove the good behaviour of the proposed interpolation function several data fitting examples and first order partial differential equations are solved the proposed numerical method is also extended to the elastostatic analysis where classic solid mechanics benchmark examples are solved copyright 2013 john wiley sons ltd

on forced oscillations in groups of interacting nonlinear systems abstract consider a periodically forced nonlinear system which can be presented as a collection of smaller subsystems with pairwise interactions between them each subsystem is assumed to be a massive point moving with friction on a compact surface possibly with a boundary in an external periodic field we present sufficient conditions for the existence of a periodic solution for the whole system the result is illustrated by a series of examples including a chain of strongly coupled pendulums in a periodic field

the distribution of the summatory function of the mobius function the summatory function of the mobius function is denoted m x in this article we deduce conditional results concerning m x assuming the riemann hypothesis and a conjecture of gonek and hejhal on the negative moments of the riemann zeta function assuming these conjectures we show that m x when appropriately normalized possesses a limiting distribution and also that a strong form of the weak mertens conjecture is true finally we speculate on the lower order of m x by studying the constructed distribution function

periodic solutions for systems of forced coupled pendulum like equations abstract the existence of periodic solutions for systems of forced pendulum like equations was studied in the papers by j a marlin internat j nonlinear mech 3 1968 439 447 and j mawhin internat j nonlinear mech 5 1970 335 339 in both works some symmetry hypotheses on the forcing terms were considered this paper discusses the existence and multiplicity of periodic solutions of systems under consideration without any requirement on the symmetry of the forcing terms note that as a model example it is possible to consider the motion of n coupled pendulums see the already mentioned paper by j a marlin or the oscillations of an n coupled point josephson junction with external time dependent disturbances studied in the autonomous case by m levi f c hoppensteadt and w l miranker quart appl math 36 1978 167 198

twisted second moments of the riemann zeta function and applications abstract in order to compute a twisted second moment of the riemann zeta function two different mollifiers each being a combination of two different dirichlet polynomials were introduced separately by bui conrey and young and by feng in this article we introduce a mollifier which is a combination of four dirichlet polynomials of different shapes we provide an asymptotic result for the twisted second moment of s for such choice of mollifier a small increment on the percentage of zeros of the riemann zeta function on the critical line is given as an application of our results

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

the twisted fourth moment of the riemann zeta function we compute the asymptotics of the fourth moment of the riemann zeta func tion times an arbitrary dirichlet polynomial of length t 1 11 the study of the moments of the riemann zeta function has a long and distinguished history starting with the work of hardy and littlewood in 1918 and continuing to the present day one motivation for understanding moments is that they yield information about the maximum size of the zeta function the lindelof hypothesis another application is to zero density estimates which in turn have consequences for primes in short intervals however they have become an interesting topic in their own right very few rigorous results are known just the second and fourth power moments indeed it is only recently that a believable conjecture for higher powers has been made the twisted moments that is moments of the riemann zeta function times an arbitrary dirichlet polynomial are important too for example levinson s method of detecting zeros of the zeta function lying on the critical line requires knowing the asymptotics of the mollified second moment in a series of papers duke friedlander and iwaniec used estimates for amplified moments of a family of l functions in order to deduce a subconvexity bound for an individual member of the family of course there are far easier methods to give a subconvexity bound for zeta but there are close analogies between different families and it is desirable to understand the structure of these amplified moments in general in this paper we prove an asymptotic formula for the twisted fourth moment of the riemann zeta function where we may take a dirichlet polynomial of length up to t 1 11 e

primes in arithmetic progressions strengthening work of rosser schoenfeld and mccurley we establish explicit chebyshev type estimates in the prime number theorem for arithmetic progressions for all moduli k 72 and other small moduli

improved error bounds for scattered data interpolation by radial basis functions if additional smoothness requirements and boundary conditions are met the well known approximation orders of scattered data interpolants by radial functions can roughly be doubled

random matrices and l functions in recent years there has been a growing interest in connections between the statistical properties of number theoretical l functions and random matrix theory we review the history of these connections some of the major achievements and a number of applications

a new bound for the smallest x with p x li x let x denote the number of primes li x this improves earlier bounds of skewes lehman and te riele we also plot more than 10000 values of x li x in four different regions including the regions discovered by lehman te riele and the authors of this paper and a more distant region in the vicinity of 1 617 x 10 9608 where x appears to exceed li x by more than 18x 1 2 log x the plots strongly suggest although upper bounds derived to date for li x x are not sufficient for a proof that x exceeds li x for at least 10 311 integers in the vicinity of 1 398 x 10 316 if it is possible to improve our bound for x li x by finding a sign change before 10 316 our first plot clearly delineates the potential candidates finally we compute the logarithmic density of li x x and find that as x departs from the region in the vicinity of 1 62 x 10 9608 the density is 1 2 7 x 10 7 99999973 and that it varies from this by no more than 9 x 10 8 over the next 10 30000 integers this should be compared to rubinstein and sarnak

zeros of combinations of the riemann ξ function and the confluent hypergeometric function on bounded vertical shifts abstract in 1914 hardy proved that infinitely many non trivial zeros of the riemann zeta function lie on the critical line using the transformation formula of the jacobi theta function recently the first author obtained an integral representation involving the riemann function and the confluent hypergeometric function linked to the general theta transformation using this result we show that a series consisting of bounded vertical shifts of a product of the riemann function and the real part of a confluent hypergeometric function has infinitely many zeros on the critical line thereby generalizing a previous result due to the first and the last authors along with roy and robles the latter itself is a generalization of hardy s theorem

the twisted mean square and critical zeros of dirichlet l functions in this work we obtain an asymptotic formula for the twisted mean square of a dirichlet l function with a longer mollifier whose coefficients are also more general than before as an application we obtain that for every dirichlet l function more than 41 72 of zeros are on the critical line and more than 40 74 of zeros are simple and on the critical line these proportions also improve previous results which were proved only for the riemann zeta function

calculation of values of l functions associated to elliptic curves we calculated numerically the values of l functions of four typical elliptic curves in the critical strip in the range im s 0 which enables us to calculate for large im s furthermore we remark that a relation exists between sato tate conjecture and the generalized riemann hypothesis