on the extreme values of the riemann zeta function on random intervals of the critical line in the present paper we show that under the riemann hypothesis and for fixed h epsilon 0 the supremum of the real and the imaginary parts of log zeta 1 2 it for t in ut h ut h are in the interval 1 epsilon log log t 1 epsilon log log t with probability tending to 1 when t goes to infinity u being a uniform random variable in 0 1 this proves a weak version of a conjecture by fyodorov hiary and keating which has recently been intensively studied in the setting of random matrices we also unconditionally show that the supremum of mathfrak r log zeta 1 2 it is at most log log t g t with probability tending to 1 g being any function tending to infinity at infinity

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

torsion homology growth and cycle complexity of arithmetic manifolds let m be an arithmetic hyperbolic 3 manifold such as a bianchi manifold we conjecture that there is a basis for the second homology of m where each basis element is represented by a surface of low genus and give evidence for this we explain the relationship between this conjecture and the study of torsion homology growth

sample size determination within the scope of conditional maximum likelihood estimation with special focus on testing the rasch model this paper refers to the exponential family of probability distributions and the conditional maximum likelihood cml theory it is concerned with the determination of the sample size for three groups of tests of linear hypotheses known as the fundamental trinity of wald score and likelihood ratio tests the main practical purpose refers to the special case of tests of the class of rasch models the theoretical background is discussed and the formal framework for sample size calculations is provided given a predetermined deviation from the model to be tested and the probabilities of the errors of the first and second kinds

on the growth of l 2 invariants for sequences of lattices in lie groups we study the asymptotic behaviour of betti numbers twisted torsion and other spectral invariants of sequences of locally symmetric spaces our main results are uniform versions of the degeorge wallach theorem of a theorem of delorme and various other limit multiplicity theorems a basic idea is to adapt the notion of benjamini schramm convergence bs convergence originally introduced for sequences of finite graphs of bounded degree to sequences of riemannian manifolds and analyze the possible limits we show that bs convergence of locally symmetric spaces g k implies convergence in an appropriate sense of the normalized relative plancherel measures associated to l2 g this then yields convergence of normalized multiplicities of unitary representations betti numbers and other spectral invariants on the other hand when the corresponding lie group g is simple and of real rank at least two we prove that there is only one possible bs limit i e when the volume tends to infinity locally symmetric spaces always bs converge to their universal cover g k this leads to various general uniform results when restricting to arbitrary sequences of congruence covers of a fixed arithmetic manifold we prove a strong quantitative version of bs convergence which in turn implies upper estimates on the rate of convergence of normalized betti numbers in the spirit of sarnak xue an important role in our approach is played by the notion of invariant random subgroups for higher rank simple lie groups g we exploit rigidity theory and in particular the nevo stuck zimmer theorem and kazhdan s property t to obtain a complete understanding of the space of irs s of g 2017 department of mathematics princeton university

bounds for the multiplicities of cohomological automorphic forms on gl2 we prove a power saving for the dimension of the space of cohomological automorphic forms of xed level and growing weight on gl 2 over any number eld that is not totally real our proof involves the theory of p adically completed cohomology developed by calegari and emerton and a bound for the growth of coinvariants in certain nitely generated noncommutative iwasawa modules

averages of characteristic polynomials in random matrix theory we compute averages of products and ratios of characteristic polynomials associated with orthogonal unitary and symplectic ensembles of random matrix theory the pfaffian determinantal formulae for these averages are obtained and the bulk scaling asymptotic limits are found for ensembles with gaussian weights classical results for the correlation functions of the random matrix ensembles and their bulk scaling limits are deduced from these formulae by a simple computation we employ a discrete approximation method the problem is solved for discrete analogues of random matrix ensembles originating from representation theory and then a limit transition is performed exact pfaffian determinantal formulae for the discrete averages are proven using standard tools of linear algebra no application of orthogonal or skew orthogonal polynomials is needed 2005 wiley periodicals inc

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 quantum unique ergodicity conjecture for thin sets abstract we consider some analogs of the quantum unique ergodicity conjecture for geodesics and shrinking families of sets in particular we prove the analog of the que conjecture for eisenstein series restricted to the infinite geodesic connecting 0 and inside the modular surface

rank combinatorial cost and homology torsion growth in higher rank lattices we investigate the rank gradient and growth of torsion in homology in residually finite groups as a tool we introduce a new complexity notion for generating sets using measured groupoids and combinatorial cost as an application we prove the vanishing of the above invariants for farber sequences of subgroups of right angled groups a group is right angled if it can be generated by a sequence of elements of infinite order such that any two consecutive elements commute most nonuniform lattices in higher rank simple lie groups are right angled we provide the first examples of uniform cocompact right angled arithmetic groups in sl n n 3 and so p q for some values of p q this is a class of lattices for which the congruence subgroup property is not known in general by using rigidity theory and the notion of invariant random subgroups it follows that both the rank gradient and the homology torsion growth vanish for an arbitrary sequence of subgroups in any right angled lattice in a higher rank simple lie group 2017

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

a rasch model for partial credit scoring a unidimensional latent trait model for responses scored in two or more ordered categories is developed this partial credit model is a member of the family of latent trait models which share the property of parameter separability and so permit specifically objective comparisons of persons and items the model can be viewed as an extension of andrich s rating scale model to situations in which ordered response alternatives are free to vary in number and structure from item to item the difference between the parameters in this model and the category boundaries in samejima s graded response model is demonstrated an unconditional maximum likelihood procedure for estimating the model parameters is developed

real zeros of quadratic dirichlet l functions a small part of the generalized riemann hypothesis asserts that l functions do not have zeros on the line segment 2 1 the question of vanishing at s 2 often has deep arithmetical significance and has been investigated extensively a persuasive view is that l functions vanish at 2 either for trivial reasons the sign of the functional equation being negative or for deep arithmetical reasons such as the l function of an elliptic curve of positive rank and that the latter case happens very rarely n katz and p sarnak 7 have formulated precise conjectures on the low lying zeros in families of l functions which support this view in the case of dirichlet l functions it is expected that l 2 is never zero and so l 0 for all 2 1 this conjecture appears to have been first enunciated by s d chowla 2 in the special case of quadratic characters progress towards these non vanishing questions has been in two directions zero density type results which establish that very few l functions have a zero in 2 1 see for example a selberg 10 m jutila 6 and d r heath brown 4 and a growing body of work on non vanishing at 2 see for example r balasubramanian and v k murty 1 h iwaniec and sarnak 5 and k soundararajan 11 further much numerical evidence for the grh has been accumulated and these calculations support chowla s conjecture see 8 and 9 however the state of knowledge could not exclude the possibility that every dirichlet l function of sufficiently large conductor has a non trivial real zero in this

large greatest common divisor sums and extreme values of the riemann zeta function it is shown that the maximum of 1 2 it on the interval t1 2 t t is at least exp 1 2 o 1 logtlogloglogt loglogt our proof uses soundararajan s resonance method and a certain large greatest common divisor sum the method of proof shows that the absolute constant a in the inequality sup 1 n1 nn k l 1ngcd nk nl nknl nexp alognlogloglognloglogn established in a recent paper of ours cannot be taken smaller than 1

lower bounds for the maximum of the riemann zeta function along vertical lines let alpha in 1 2 1 be fixed we prove that begin aligned max 0 le t le t zeta alpha it ge exp left frac c alpha log t 1 alpha log log t alpha right end aligned for all sufficiently large t where we can choose c alpha 0 18 2 alpha 1 1 alpha the same result has already been obtained by montgomery with a smaller value for c alpha however our proof which uses a modified version of soundararajan s resonance method together with ideas of hilberdink is completely different from montgomery s this new proof also allows us to obtain lower bounds for the measure of those t in 0 t for which zeta alpha it is of the order mentioned above

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

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

maximum of the riemann zeta function on a short interval of the critical line we prove the leading order of a conjecture by fyodorov hiary and keating about the maximum of the riemann zeta function on random intervals along the critical line more precisely as t for a set of t t 2t of measure 1 o 1 t we have max t u 1log 12 iu 1 o 1 loglogt

the strong multiplicity one theorem for text gl _n let a be a global field and denote by a its ring of adeles let 0 be a unitary character of a x trivial on k for the group g gln let c l ga gfc 0 be the space of cusp forms on g with central character the multiplicity one theorem first proved by jacquet langlands in 2 for n 2 and by shalika in 11 for all n states that each irreducible unitary representation of g occurs with multiplicity at most one in c it is also well known that each irreducible unitary representation of g occurring in c has a factorization 7r v 7rv where each ttv is a representation of the local group gv gln kv as a complement to the multiplicity one theorem one has the following rigidity theorem see 9 p 209 4 p 552 for n 2 1 p 307 7 p 187

power moments and value distribution of functions in this paper we study various abscissae which one can associate to a given function f or rather to the power moments of f these are motivated by long standing open problems in analytic number theory we show how these abscissae connect to the distribution of values of f in a very elegant way using convex conjugates this connection allows us to show which abscissae are realizable for both general and more specific arithmetical functions further it may give a new approach to for example dirichlet s divisor problem

on the averages of characteristic polynomials from classical groups we provide an elementary and self contained derivation of formulae for averages of products and ratios of characteristic polynomials of random matrices from classical groups using classical results due to weyl and littlewood

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

upper bounds for lq norms of dirichlet polynomials with small q abstract we improve on previous upper bounds for the qth norm of the partial sums of the riemann zeta function on the half line when 0 q 1 in particular we show that the 1 norm is bounded above by log n 1 4 log log n 1 4

quantum unique ergodicity for half integral weight automorphic forms we investigate the analogue of the quantum unique ergodicity que conjecture for half integral weight automorphic forms assuming the generalized riemann hypothesis grh we establish que for both half integral weight holomorphic hecke cusp forms for 0 4 lying in kohnen s plus subspace and for half integral weight hecke maa cusp forms for 0 4 lying in kohnen s plus subspace by combining the former result along with an argument of rudnick it follows that under grh the zeros of these holomorphic hecke cusp equidistribute with respect to hyperbolic measure on 0 4 h as the weight tends to infinity

strong szegő asymptotics and zeros of the zeta function assuming the riemann hypothesis we prove the weak convergence of linear statistics of the zeros of the riemann function to a gaussian field with covariance structure corresponding to the h1 2 norm of the test functions for this purpose we obtain an approximate form of the explicit formula relying on selberg s smoothed expression for and the helffer sjostrand functional calculus our main result is an analogue of the strong szeg theorem known for toeplitz operators and random matrix theory 2014 wiley periodicals inc

coefficient of maass forms and the siegel zero with an appendix by dorian goldfeld jeffrey hoffstein and daniel lieman

moments and distribution of central l values of quadratic twists of elliptic curves we show that if one can compute a little more than a particular moment for some family of l functions then one has upper bounds of the conjectured order of magnitude for all smaller positive real moments and a one sided central limit theorem holds we illustrate our method for the family of quadratic twists of an elliptic curve obtaining sharp upper bounds for all moments below the first we also establish a one sided central limit theorem supporting a conjecture of keating and snaith our work leads to a conjecture on the distribution of the order of the tate shafarevich group for rank zero quadratic twists of an elliptic curve and establishes the upper bound part of this conjecture assuming the birch swinnerton dyer conjecture

number variance for arithmetic hyperbolic surfaces we prove that the number variance for the spectrum of an arithmetic surface is highly nonrigid in part of the universal range in fact it is close to having a poisson behavior this fact was discovered numerically by schmit bogomolny georgeot and giannoni it has its origin in the high degeneracy of the length spectrum first observed by selberg

average ranks of elliptic curves tension between data and conjecture rational points on elliptic curves are the gems of arithmetic they are to diophantine geometry what units in rings of integers are to algebraic number theory what algebraic cycles are to algebraic geometry a rational point in just the right context at one place in the theory can inhibit and control thanks to ideas of kolyvagin kol88 the existence of rational points and other mathematical structures elsewhere despite all that we know about these objects the initial mystery and excitement that drew mathematicians to this arena in the first place remains in full force today we have a network of heuristics and conjectures regarding rational points and we have massive data accumulated to exhibit instances of the phenomena generally we would expect that our data support our conjectures and if not we lose faith in our conjectures but here there is a somewhat more surprising interrelation between data and conjecture they are not exactly in open conflict one with the other but they are no great comfort to each other either we discuss various aspects of this story including recent heuristics and data that attempt to resolve this mystery we shall try to convince the reader that despite seeming discrepancy data and conjecture are in fact in harmony

multiplicity one theorems in the local characteristic 0 non archimedean case we consider distributions on gl n 1 which are invariant under the adjoint action of gl n we prove that such distributions are invariant by transposition this implies multiplicity at most one for restrictions from gl n 1 to gl n similar theorems are obtained for orthogonal or unitary groups

extreme values of the riemann zeta function and its argument we combine our version of the resonance method with certain convolution formulas for zeta s and log zeta s this leads to a new omega result for zeta 1 2 it the maximum of zeta 1 2 it on the interval 1 le t le t is at least exp left 1 o 1 sqrt log t log log log t log log t right we also obtain conditional results for s t 1 pi times the argument of zeta 1 2 it and s 1 t int 0 t s tau d tau on the riemann hypothesis the maximum of s t is at least c sqrt log t log log log t log log t and the maximum of s 1 t is at least c 1 sqrt log t log log log t log log t 3 on the interval t beta le t le t whenever 0 le beta 1

the maximum size of l functions we conjecture the true rate of growth of the maximum size of the riemann zeta function and other l functions we support our conjecture using arguments from random matrix theory conjectures for moments of l functions and also by assuming a random model for the primes

linear statistics of point processes via orthogonal polynomials for arbitrary 0 we use the orthogonal polynomials techniques developed in killip and nenciu in arxiv math 0508113v1 2005 killip and nenciu in int math res not 50 2665 2701 2004 to study certain linear statistics associated with the circular and jacobi ensembles we identify the distribution of these statistics then prove a joint central limit theorem in the circular case similar statements have been proved using different methods by a number of authors in the jacobi case these results are new

an extension of the rasch model for ratings providing both location and dispersion parameters an elaboration of a psychometric model for rated data which belongs to the class of rasch models is shown to provide a model with two parameters one characterising location and one characterising dispersion the later parameter derived from the idea of a unit of scale is also shown to reflect the shape of rating distributions ranging from unimodal through uniform and then to u shaped distributions a brief case is made that when a rating distribution is treated as a random error distribution then the distribution should be unimodal

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

on the sl 2 period integral let e f be a quadratic extension of number fields for a cuspidal representation of sl2 ae we study in this paper the integral of functions in on sl2 f sl2 af we characterize the nonvanishing of these integrals called period integrals in terms of having a whittaker model with respect to characters of e ae which are trivial on af we show that the period integral in general is not a product of local invariant functionals and find a necessary and sufficient condition when it is we exhibit cuspidal representations of sl2 ae whose period integral vanishes identically while each local constituent admits an sl2 invariant linear functional finally we construct an automorphic representation on sl2 ae which is abstractly sl2 af distinguished but for which none of the elements in the global l packet determined by it is distinguished by sl2 af

effective multiplicity one on gl n and narrow zero free regions for rankin selberg l functions we establish zero free regions tapering as an inverse power of the analytic conductor for rankin selberg l functions on gln gln such zero free regions are equivalent to commensurate lower bounds on the edge of the critical strip and in the case of l s on the residue at s 1 as an application we show that a cuspidal automorphic representation on gln is determined by a finite number of its dirichlet series coefficients and that this number grows at most polynomially in the analytic conductor introduction let a be the ring of adeles over a number field f and let and be two cuspidal representations of gln a with restricted tensor product decompositions v v and v

mesoscopic fluctuations of the zeta zeros we prove a multidimensional extension of selberg s central limit theorem for log in which non trivial correlations appear in particular this answers a question by coram and diaconis about the mesoscopic fluctuations of the zeros of the riemann zeta function similar results are given in the context of random matrices from the unitary group this shows the correspondence n log t not only between the dimension of the matrix and the height on the critical line but also in a local scale for small deviations from the critical axis or the unit circle

periods of cusp forms and elliptic curves over imaginary quadratic fields in this paper we explore the arithmetic correspondence between on the one hand isogeny classes of elliptic curves e defined over an imaginary quadratic field k of class number one and on the other hand rational newforms f of weight two for the congruence subgroups where n is an ideal in the ring of integers r of k this continues work of the first author and forms part of the ph d thesis of the second author in each case we compute numerically the value of the l series at and compare with the value of which is predicted by the birch swinnerton dyer conjecture finding agreement to several decimal places in particular we find that whenever has a point of infinite order several examples are given in detail from the extensive tables computed by the authors

value distribution of l ρ abstract let l be an automorphic l function assuming the riemann hypothesis for l s and the selberg normality conjecture we obtain a lower bound for the second negative moment and extreme small values of l where is a zero of l s

arithmetic zeros and nodal domains on the sphere we obtain lower bounds for the number of nodal domains of hecke eigenfunctions on the sphere assuming the generalized lindelof hypothesis we prove that the number of nodal domains of any hecke eigenfunction grows with the eigenvalue of the laplacian by a very different method we show unconditionally that the average number of nodal domains of degree l hecke eigenfunctions grows significantly faster than the uniform growth obtained under lindelof

developments in random matrix theory in this introduction to the journal of physics a special issue on random matrix theory we give a review of the main historical developments in random matrix theory a short summary of the papers that appear in this special issue is also given

the eighth moment of dirichlet l functions abstract we prove an asymptotic for the eighth moment of dirichlet l functions averaged over primitive characters modulo q over all moduli q q and with a short average on the critical line conditionally on grh we derive the analogous result for the fourth moment of dirichlet twists of gl 2 l functions our results match the moment conjectures in the literature in particular the constant 24 024 appears as a factor in the leading order term of the eighth moment

exponential growth of torsion in the cohomology of arithmetic hyperbolic manifolds for d 2n 1 a positive odd integer we consider sequences of arithmetic subgroups of text so 0 d 1 and yielding corresponding hyperbolic manifolds of finite volume and show that under appropriate and natural assumptions the torsion of the associated cohomology groups grows exponentially

moments of the logarithmic derivative of characteristic polynomials from so n and usp 2n we study moments of the logarithmic derivative of characteristic polynomials of orthogonal and symplectic random matrices in particular we compute the asymptotics for large matrix size n of these moments evaluated at points that are approaching 1 this follows the work of bailey et al j math phys 60 8 083509 2019 where they computed these asymptotics in the case of unitary random matrices

the fourth moment of quadratic dirichlet l functions we study the fourth moment of quadratic dirichlet l functions at s frac 1 2 we show an asymptotic formula under the generalized riemann hypothesis and obtain a precise lower bound unconditionally the proofs of these results follow closely arguments of soundararajan and young j eur math soc 12 5 1097 1116 2010 and soundararajan ann math 2 152 2 447 488 2000

moments of the riemann zeta function assuming the riemann hypothesis we obtain an upper bound for the moments of the riemann zeta function on the critical line our bound is nearly as sharp as the conjectured asymptotic formulae for these moments the method extends to moments in other families of l functions

arithmetical chaos and violation of universality in energy level statistics a class of strongly chaotic systems revealing a strange arithmetical structure is discussed whose quantal energy levels exhibit level attraction rather than repulsion as an example the nearest neighbor level spacings for artin s billiard have been computed in a large energy range it is shown that the observed violation of universality has its root in the existence of an infinite number of hermitian operators hecke operators which commute with the hamiltonian and generate nongeneric correlations in the eigenfunctions

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 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

landau siegel zeros and zeros of the derivative of the riemann zeta function abstract we show that if the derivative of the riemann zeta function has sufficiently many zeros close to the critical line then the zeta function has many closely spaced zeros this gives a condition on the zeros of the derivative of the zeta function which implies a lower bound of the class numbers of imaginary quadratic fields

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

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

norms on the cohomology of hyperbolic 3 manifolds we study the relationship between two norms on the first cohomology of a hyperbolic 3 manifold the purely topological thurston norm and the more geometric harmonic norm refining recent results of bergeron engun and venkatesh as well as older work of kronheimer and mrowka we show that these norms are roughly proportional with explicit constants depending only on the volume and injectivity radius of the hyperbolic 3 manifold itself moreover we give families of examples showing that some but not all qualitative aspects of our estimates are sharp finally we exhibit closed hyperbolic 3 manifolds where the thurston norm grows exponentially in terms of the volume and yet there is a uniform lower bound on the injectivity radius

refinements of gal s theorem and applications abstract we give a simple proof of a well known theorem of gal and of the recent related results of aistleitner berkes and seip 1 regarding the size of gcd sums in fact our method obtains the asymptotically sharp constant in gal s theorem which is new our approach also gives a transparent explanation of the relationship between the maximal size of the riemann zeta function on vertical lines and bounds on gcd sums a point which was previously unclear furthermore we obtain sharp bounds on the spectral norm of gcd matrices which settles a question raised in 2 we use bounds for the spectral norm to show that series formed out of dilates of periodic functions of bounded variation converge almost everywhere if the coefficients of the series are in l 2 log log 1 l with 2 this was previously known with 4 and is known to fail for 2 we also develop a sharp carleson hunt type theorem for functions of bounded variations which settles another question raised in 1 finally we obtain almost sure bounds for partial sums of dilates of periodic functions of bounded variations improving 1

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

extreme values of zeta and l functions we introduce a resonance method to produce large values of the riemann zeta function on the critical line and large and small central values of l functions

sums of divisor functions in fq t and matrix integrals we study the mean square of sums of the kth divisor function d k n over short intervals and arithmetic progressions for the rational function field over a finite field of q elements in the limit as q rightarrow infty we establish a relationship with a matrix integral over the unitary group evaluating this integral enables us to compute the mean square of the sums of d k n in terms of a lattice point count this lattice point count can in turn be calculated in terms of a certain piecewise polynomial function which we analyse our results suggest general conjectures for the corresponding classical problems over the integers which agree with the few cases where the answer is known

sparse equidistribution problems period bounds and subconvexity we introduce a geometric method to bound periods of automorphic forms the key features of this method are the use of equidistribution results in place of mean value theorems and the systematic use of mixing and the spectral gap applications are given to equidistribution of sparse subsets of horocycles and to equidistribution of cm points to subconvexity of the triple product period in the level aspect over number fields which implies subconvexity for certain standard and rankin selberg l functions and to bounding fourier coefficients of automorphic forms

large values of l functions from the selberg class abstract in the present paper we prove lower bounds for l functions from the selberg class by this means improving earlier results obtained by the second author together with jorn steuding we formulate two theorems which use slightly different technical assumptions and give two totally different proofs the first proof uses the resonance method which was introduced by soundararajan while the second proof uses methods from diophantine approximation which resemble those used by montgomery interestingly both methods lead to roughly the same lower bounds which fall short of those known for the riemann zeta function and seem to be difficult to be improved additionally to these results we also prove upper bounds for l functions in the selberg class and present a further application of a theorem of chen which is used in the diophantine approximation method mentioned above

further remarks on nondichotomization of graded responses i have tried to indicate not only the analytic differences in some of the points made in the paper and the rejoinder but also the perspective from which these differences arise the point of the paper and these remarks is to show how the special case of the mpm revealed why the joining assumption and dichotomization are not in general properties of the urm for graded responses and thereby to identify the circumstances where one would require that this property did not hold in empirical graded responses

lower bounds for moments of l functions the moments of central values of families of l functions have recently attracted much attention and with the work of keating and snaith 2000 commun math phys 214 57 89 and 91 110 there are now precise conjectures for their limiting values we develop a simple method to establish lower bounds of the conjectured order of magnitude for several such families of l functions as an example we work out the case of the family of all dirichlet l functions to a prime modulus

bounds for multiplicities of unitary representations of cohomological type in spaces of cusp forms let goo be a semisimple real lie group with unitary dual goo we produce new upper bounds for the multiplicities with which representations e of cohomological type appear in certain spaces of cusp forms on goo the main new idea is to apply noncommutative iwasawa theory to certain p adic completions of the cohomology of locally symmetric spaces

a rasch model for continuous ratings a unidimensional latent trait model for continuous ratings is developed this model is an extension of andrich s rating formulation which assumes that the response process at latent thresholds is governed by the dichotomous rasch model item characteristic functions and information functions are used to illustrate that the model takes ceiling and floor effects into account both the dichotomous rasch model and a linear model with normally distributed error can be derived as limiting cases the separability of the structural and incidental parameters is demonstrated and a procedure for estimating the parameters is outlined