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

joint universality theorems for pairs of automorphic zeta functions in this paper we establish the joint universality theorems for pairs consisting of the riemann zeta function automorphic l functions associated with holomorphic hecke eigen cusp forms rankin selberg l functions and symmetric square l functions

selberg s orthonormality conjecture and joint universality of l functions in the paper we introduce a new method how to use only an orthonormality relation of coefficients of dirichlet series defining given l functions from the selberg class to prove joint universality

necessary and sufficient conditions for doubly stochasticity of infinite dimensional quadratic operators abstract we define a doubly stochastic quadratic operator on infinite dimensional space we give necessary and sufficient conditions for doubly stochasticity of a quadratic operator

controllable fuzzy matrices abstract controllable fuzzy matrices are defined several properties of controllable fuzzy matrices and nilpotent fuzzy matrices are studied we obtain an algorithm to judge whether a fuzzy matrix is controllable and reduce a controllable fuzzy matrix to canonical form the results presented generalize analogous results concerning strongly transitive fuzzy matrices and nilpotent fuzzy matrices known in the literature

random matrix theory and l functions at s 1 2 recent results of katz and sarnak 8 9 suggest that the low lying zeros of families of l functions display the statistics of the eigenvalues of one of the compact groups of matrices u n o n or usp 2n we here explore the link between the value distributions of the l functions within these families at the central point s 1 2 and those of the characteristic polynomials z u of matrices u with respect to averages over so 2n and usp 2n at the corresponding point 0 using techniques previously developed for u n in 10 for any matrix size n we find exact expressions for the moments of z u 0 for each ensemble and hence calculate the asymptotic large n value distributions for z u 0 and log z u 0 the asymptotic results for the integer moments agree precisely with the few corresponding values known for l functions the value distributions suggest consequences for the non vanishing of l functions at the central point

the determinant and adjoint of a square fuzzy matrix abstract the concept of the determinant theory and the adjoint of a square fuzzy matrix will be studied also we define the circular fuzzy matrices and show that some properties of a square fuzzy matrix such as reflexivity transitivity and circularity are carried out to the adjoint of the matrix finally we show how to construct a transitive fuzzy matrix from a given one through the adjoint matrix of it

numerical computations concerning the erh this paper describes a computation which established the erh to height 10000 for all primitive dirichlet l series with conductor q 13 and to height 2500 for all q 72 all composite q 112 and other moduli the computations were based on euler maclaurin summation care was taken to obtain mathematically rigorous results the zeros were first located within 10 12 then rigorously separated using an interval arithmetic package a generalized turing criterion was used to show there were no zeros off the critical line statistics about the spacings between zeros were compiled to test the pair correlation conjecture and gue hypothesis

determinant theory for fuzzy matrices abstract we study determinant theory for fuzzy square matrices including some boolean matrices and consider the cayley hamilton theorem for these matrices we prove that det a adj a det a det adj a a where adj a denotes the adjoint matrix of a fuzzy matrix a

a random matrix model for elliptic curve l functions of finite conductor we propose a random matrix model for families of elliptic curve l functions of finite conductor a repulsion of the critical zeros of these l functions away from the centre of the critical strip was observed numerically by miller 2006 exp math 15 257 79 such behaviour deviates qualitatively from the conjectural limiting distribution of the zeros for large conductors this distribution is expected to approach the one level density of eigenvalues of orthogonal matrices after appropriate rescaling our purpose here is to provide a random matrix model for miller s surprising discovery we consider the family of even quadratic twists of a given elliptic curve the main ingredient in our model is a calculation of the eigenvalue distribution of random orthogonal matrices whose characteristic polynomials are larger than some given value at the symmetry point in the spectra we call this sub ensemble of so 2n the excised orthogonal ensemble the sieving off of matrices with small values of the characteristic polynomial is akin to the discretization of the central values of l functions implied by the formulae of waldspurger and kohnen zagier the cut off scale appropriate to modelling elliptic curve l functions is exponentially small relative to the matrix size n the one level density of the excised ensemble can be expressed in terms of that of the well known jacobi ensemble enabling the former to be explicitly calculated it exhibits an exponentially small on the scale of the mean spacing hard gap determined by the cut off value followed by soft repulsion on a much larger scale neither of these features is present in the one level density of so 2n when n we recover the limiting orthogonal behaviour our results agree qualitatively with miller s discrepancy choosing the cut off appropriately gives a model in good quantitative agreement with the number theoretical data

on the mod gaussian convergence of a sum over primes we prove mod gaussian convergence for a dirichlet polynomial which approximates mathrm im log zeta 1 2 it this dirichlet polynomial is sufficiently long to deduce selberg s central limit theorem with an explicit error term moreover assuming the riemann hypothesis we apply the theory of the riemann zeta function to extend this mod gaussian convergence to the complex plane from this we obtain that mathrm im log zeta 1 2 it satisfies a large deviation principle on the critical line results about the moments of the riemann zeta function follow

demography of cirsium vulgare in a grazing experiment the complete demographies of 16 populations of cirsium vulgare were followed in a replicated experiment the experiment was a factorial combination of two intensities of sheep grazing in each of three seasons winter grazed or ungrazed spring grazed or ungrazed and summer light or heavy grazing giving eight treatments in two blocks for 6 years from 1987 to 1992 the population sizes of c vulgare were monitored in each of the 16 paddocks after 1989 grazing in spring or winter or increased grazing in summer all increased population sizes population sizes fluctuated widely between years the effects of the grazing treatments and plant sizes on the transitions between nine life history stages were determined

doubly stochastic quadratic operators and birkhoff s problem abstract in the present we introduce a concept of doubly stochastic quadratic operator we study necessary and sufficient conditions for doubly stochasticity of operator moreover we study analogue of birkhoff s theorem for the class of doubly stochastic operators

a hybrid euler hadamard product for the riemann zeta function we use a smoothed version of the explicit formula to find an accurate pointwise approximation to the riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes we model the first product by characteristic polynomials of random matrices this provides a statistical model of the zeta function which involves the primes in a natural way we then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line for the second and fourth moments we establish all of the steps in our approach rigorously this calculation illuminates recent conjectures for these moments based on connections with random matrix theory

majorization doubly stochastic matrices and comparison of eigenvalues abstract 1 basic properties of majorization 2 isotone maps and algebraic operations 3 double sub and superstochasticity 4 doubly stochastic matrices 5 doubly stochastic matrices with minimum permanent 6 comparison of eigenvalues 7 doubly stochastic maps

moments of the characteristic polynomial in the three ensembles of random matrices moments of the characteristic polynomial of a random matrix taken from any of the three ensembles orthogonal unitary or symplectic are given either as a determinant or a pfaffian or as a sum of determinants for gaussian ensembles on comparing two expressions of the same moment one obtains two remarkable identities one between an n n determinant and an m m determinant and another between the pfaffian of a 2n 2n anti symmetric matrix and a sum of m m determinants

on the location of the zero free half plane of a random epstein zeta function in this note we study for a random lattice l of large dimension n the supremum of the real parts of the zeros of the epstein zeta function e n l s and prove that this random variable scaled by n 1 has a limit distribution which we give explicitly this limit distribution is studied in some detail in particular we give an explicit formula for its distribution function furthermore we obtain a limit distribution for the frequency of zeros of e n l s in vertical strips contained in the half plane mathfrak r s frac n 2

integral moments of l functions we give a new heuristic for all of the main terms in the integral moments of various families of primitive l functions the results agree with previous conjectures for the leading order terms our conjectures also have an almost identical form to exact expressions for the corresponding moments of the characteristic polynomials of either unitary orthogonal or symplectic matrices where the moments are defined by the appropriate group averages this lends support to the idea that arithmetical l functions have a spectral interpretation and that their value distributions can be modelled using random matrix theory numerical examples show good agreement with our conjectures

some remarks on the convergence of the dirichlet series of l functions and related questions first we show that the abscissae of uniform and absolute convergence of dirichlet series coincide in the case of l functions from the selberg class mathcal s we also study the latter abscissa inside the extended selberg class indicating a different behavior in the two classes next we address two questions about majorants of functions in mathcal s showing links with the distribution of the zeros and with independence results

symmetry transitions in random matrix theory l functions we compute the moments of the characteristic polynomials of random orthogonal and symplectic matrices defined by averages with respect to haar measure on so 2n and usp 2n to leading order as n on the unit circle as functions of the angle measured from one of the two symmetry points in the eigenvalue spectrum exp pm i theta n 1 le n le n our results extend previous formulae that relate just to the symmetry points i e to 0 local spectral statistics are expected to converge to those of random unitary matrices in the limit as n when is fixed and to show a transition from the orthogonal or symplectic to the unitary forms on the scale of the mean eigenvalue spacing if y n they become functions of y in the limit when n we verify that this is true for the spectral two point correlation function but show that it is not true for the moments of the characteristic polynomials for which the leading order asymptotic approximation is a function of rather than y symmetry points therefore influence the moments asymptotically far away on the scale of the mean eigenvalue spacing we also investigate the moments of the logarithms of the characteristic polynomials in the same context the moments of the characteristic polynomials of random matrices are conjectured to be related to the moments of families of l functions previously moments at the symmetry point 0 have been related to the moments of families of l functions evaluated at the centre of the critical strip our results motivate general conjectures for the moments of orthogonal and symplectic families of l functions evaluated at a fixed height t up the critical line these conjectures suggest that the symmetry of the non trivial zeros of the l functions influences the moments asymptotically far on the scale of the mean zero spacing from the centre of the critical strip we verify that the second moments of real quadratic dirichlet l functions and a family of automorphic l functions are consistent with our conjectures

the semiring of matrices over a finite chain let l m denote the chain 0 1 2 m 1 with the usual ordering and m n l m the matrix semiring of all nxn matrices with elements in l m we firstly introduce some order preserving semiring homomorphisms from m n l m to m l k by using these homomorphisms we show that a matrix over the finite chain l m can be decomposed into the sum of some matrices over the finite chain l k where k

on the mean values of dirichlet l functions we study the 2kth power moment of dirichlet l functions l s at the centre of the critical strip s 2 where the average is over all primitive characters mod q we extend to this case the hybrid euler hadamard product results of gonek hughes and keating for the riemann zeta function this allows us to recover conjectures for the moments based on random matrix models incorporating the arithmetical terms in a natural way

random matrix theory and ζ 1 2 it we study the characteristic polynomials z u of matrices u in the circular unitary ensemble cue of random matrix theory exact expressions for any matrix size n are derived for the moments of z and z z and from these we obtain the asymptotics of the value distributions and cumulants of the real and imaginary parts of log z as n in the limit we show that these two distributions are independent and gaussian costin and lebowitz 15 previously found the gaussian limit distribution for im log z using a different approach and our result for the cumulants proves a conjecture made by them in this case we also calculate the leading order n asymptotics of the moments of z and z z these cue results are then compared with what is known about the riemann zeta function s on its critical line re s 1 2 assuming the riemann hypothesis equating the mean density of the non trivial zeros of the zeta function at a height t up the critical line with the mean density of the matrix eigenvalues gives a connection between n and t invoking this connection our cue results coincide with a theorem of selberg for the value distribution of log 1 2 it in the limit t they are also in close agreement with numerical data computed by odlyzko 29 for large but finite t this leads us to a conjecture for the moments of 1 2 it finally we generalize our random matrix results to the circular orthogonal coe and circular symplectic cse ensembles