the riemann hilbert approach to strong asymptotics for orthogonal polynomials on 1 1 abstract we consider polynomials that are orthogonal on 1 1 with respect to a modified jacobi weight 1 x 1 x h x with 1 and h real analytic and strictly positive on 1 1 we obtain full asymptotic expansions for the monic and orthonormal polynomials outside the interval 1 1 for the recurrence coefficients and for the leading coefficients of the orthonormal polynomials we also deduce asymptotic behavior for the hankel determinants and for the monic orthogonal polynomials on the interval 1 1 for the asymptotic analysis we use the steepest descent technique for riemann hilbert problems developed by deift and zhou and applied to orthogonal polynomials on the real line by deift kriecherbauer mclaughlin venakides and zhou in the steepest descent method we will use the szeg function associated with the weight and for the local analysis around the endpoints 1 we use bessel functions of appropriate order whereas deift et al use airy functions

applications and generalizations of fisher hartwig asymptotics fisher hartwig asymptotics refers to the large n form of a class of toeplitz determinants with singular generating functions this class of toeplitz determinants occurs in the study of the spin spin correlations for the two dimensional ising model and the ground state density matrix of the impenetrable bose gas amongst other problems in mathematical physics we give a new application of the original fisher hartwig formula to the asymptotic decay of the ising correlations above tc while the study of the bose gas density matrix leads us to generalize the fisher hartwig formula to the asymptotic form of random matrix averages over the classical groups and the gaussian and laguerre unitary matrix ensembles another viewpoint of our generalizations is that they extend to hankel determinants the fisher hartwig asymptotic form known for toeplitz determinants

moments of the position of the maximum for gue characteristic polynomials and for log correlated gaussian processes we study three instances of log correlated processes on the interval the logarithm of the gaussian unitary ensemble gue characteristic polynomial the gaussian log correlated potential in presence of edge charges and the fractional brownian motion with hurst index h rightarrow 0 fbm0 in previous collaborations we obtained the probability distribution function pdf of the value of the global minimum equivalently maximum for the first two processes using the freezing duality conjecture fdc here we study the pdf of the position of the maximum x m through its moments using replica this requires calculating moments of the density of eigenvalues in the beta jacobi ensemble using jack polynomials we obtain an exact and explicit expression for both positive and negative integer moments for arbitrary beta 0 and positive integer n in terms of sums over partitions for positive moments this expression agrees with a very recent independent derivation by mezzadri and reynolds we check our results against a contour integral formula derived recently by borodin and gorin presented in the appendix 1 from these authors the duality necessary for the fdc to work is proved and on our expressions found to correspond to exchange of partitions with their dual performing the limit n rightarrow 0 and to negative dyson index beta rightarrow 2 we obtain the moments of x m and give explicit expressions for the lowest ones numerical checks for the gue polynomials performed independently by n simm indicate encouraging agreement some results are also obtained for moments in laguerre hermite gaussian as well as circular and related ensembles the correlations of the position and the value of the field at the minimum are also analyzed

universal behavior for averages of characteristic polynomials at the origin of the spectrum it has been shown by strahov and fyodorov that averages of products and ratios of characteristic polynomials corresponding to hermitian matrices of a unitary ensemble involve kernels related to orthogonal polynomials and their cauchy transforms we will show that for the unitary ensemble open image in new window hermitian matrices these kernels have universal behavior at the origin of the spectrum as n in terms of bessel functions our approach is based on the characterization of orthogonal polynomials together with their cauchy transforms via a matrix riemann hilbert problem due to fokas its and kitaev and on an application of the deift zhou steepest descent method for matrix riemann hilbert problems to obtain the asymptotic behavior of the riemann hilbert problem

random hermitian matrices and gaussian multiplicative chaos n berestycki s work is supported by epsrc grants ep l018896 1 and ep i03372x 1 m d wong is a phd student at the cambridge centre for analysis supported by epsrc grant ep l016516 1 some of this work was carried out while the first and third authors visited the university of helsinki funded in part by epsrc grant ep l018896 1 they also wish to thank the university of helsinki for its hospitality during this visit c webb wishes to thank the isaac newton institute for mathematical sciences for its hospitality during the random geometry program during which this project was initiated c webb was supported by the eemil aaltonen foundation grant stochastic dynamics on large random graphs and academy of finland grants 288318 and 308123

simultaneous reduction of sets of matrices under similarity abstract let s be a set of n n matrices over a field f and a the algebra generated by s over f the problem of deciding whether the elements of s can be simultaneously reduced to block triangular form with the diagonal blocks of some specified size is considered and an account is given of various methods used to attack the problem most of the techniques use representation theory to obtain information on a the problems of simultaneous triangularization existence of common eigenvectors etc are also considered the aim of the paper is to survey the methods used to attack these problems and to give some typical results the paper does not contain many new results

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

sums of kloosterman sums in arithmetic progressions and the error term in the dispersion method we prove a bound for quintilinear sums of kloosterman sums with congruence conditions on the smooth summation variables this generalizes classical work of deshouillers and iwaniec and is key to obtaining power saving error terms in applications notably the dispersion method as a consequence assuming the riemann hypothesis for dirichlet l functions we prove a power saving error term in the titchmarsh divisor problem of estimating sum p leq x tau p 1 unconditionally we isolate the possible contribution of siegel zeroes showing it is always negative extending work of fouvry and tenenbaum we obtain power saving in the asymptotic formula for sum n leq x tau k n tau n 1 reproving a result announced by bykovski u i and vinogradov by a different method the gain in the exponent is shown to be independent of k if a generalized lindel of hypothesis is assumed

on moments of twisted l functions we study the average of the product of the central values of two l functions of modular forms f and g twisted by dirichlet characters to a large prime modulus q as our principal tools we use spectral theory to develop bounds on averages of shifted convolution sums with differences ranging over multiples of q and we use the theory of deligne and katz to prove new bounds on bilinear forms in kloosterman sums with power savings when both variables are near the square root of q when at least one of the forms f and g is non cuspidal we obtain an asymptotic formula for the mixed second moment of twisted l functions with a power saving error term in particular when both are non cuspidal this gives a significant improvement on m young s asymptotic evaluation of the fourth moment of dirichlet l functions in the general case the asymptotic formula with a power saving is proved under a conjectural estimate for certain bilinear forms in kloosterman sums

l functions for symmetric products of kloosterman sums the classical kloosterman sums give rise to a galois representation of the function field unramified outside 0 and we study the local monodromy of this representation at using l adic method based on the work of deligne and katz as an application we determine the degrees and the bad factors of the l functions of the symmetric products of the above representation our results generalize some results of robba obtained through p adic method

random matrix theory the exceptional lie groups and l functions there has recently been interest in relating properties of matrices drawn at random from the classical compact groups to statistical characteristics of number theoretical l functions one example is the relationship conjectured to hold between the value distributions of the characteristic polynomials of such matrices and value distributions within families of l functions these connections are extended here to non classical groups we focus on an explicit example the exceptional lie group g2 the value distributions for characteristic polynomials associated with the 7 and 14 dimensional representations of g2 defined with respect to the uniform invariant haar measure are calculated using two of the macdonald constant term identities a one parameter family of l functions over a finite field is described whose value distribution in the limit as the size of the finite field grows is related to that of the characteristic polynomials associated with the seven dimensional representation of g2 the random matrix calculations extend to all exceptional lie groups

the isomonodromy approach to matric models in 2d quantum gravity we consider the double scaling limit in the hermitian matrix model for 2d quantum gravity associated with the measure exp sum limits j 1 n t j z 2j n geqq 3 we show that after the appropriate modification of the contour of integration the cross migdal douglas shenker limit to the painleve i equation in the generic case of the pure gravity is valid and calculate the nonperturbative parameters of the corresponding painleve function our approach is based on the wkb analysis of the l a pair corresponding to the discrete string equation in the framework of the inverse monodromy method here we extend our results which were obtained before for the particular casesn 2 3 our analysis complements the isomonodromy approach proposed by g moore to the general string equations that come from the matrix model in the continuous limit and differ in that we apply the isomonodromy technique to investigate the double scaling limit itself

algebras generated by two idempotents abstract it is shown that a noncommutative simple algebra generated over a field f by two idempotents is necessarily the ring of 2 2 matrices over a simple extension of f and that every matrix ring over a field k can be generated over k by three idempotents

applications of the kuznetsov formula on gl 3 we develop a fairly explicit kuznetsov formula on gl 3 and discuss the analytic behavior of the test functions on both sides applications to weyl s law exceptional eigenvalues a large sieve and l functions are given

hankel determinant and orthogonal polynomials for a gaussian weight with a discontinuity at the edge we compute asymptotics for hankel determinants and orthogonal polynomials with respect to a discontinuous gaussian weight in a critical regime where the discontinuity is close to the edge of the associated equilibrium measure support their behavior is described in terms of the ablowitz segur family of solutions to the painleve ii equation our results complement the ones in 33 as consequences of our results we conjecture asymptotics for an airy kernel fredholm determinant and total integral identities for painleve ii transcendents and we also prove a new result on the poles of the ablowitz segur solutions to the painleve ii equation we also highlight applications of our results in random matrix theory

pairs of matrices with quadratic minimal polynomials abstract let a b be n n matrices over a field f and suppose a b have quadratic minimal polynomials then the algebra generated by a b has dimension at most 2n 1 if n is odd and 2n if n is even these bounds are exact

trilinear forms with kloosterman fractions abstract we give new bounds for a m n m n a e a m n where m n and a are arbitrary coefficients improving upon a result of duke friedlander and iwaniec 7 we also apply these bounds to problems on representations by determinant equations and on the equidistribution of solutions to linear equations

algebraic trace functions over the primes we study sums over primes of trace functions of l adic sheaves using an extension of our earlier results on algebraic twists of modular forms to the case of eisenstein series and bounds for type ii sums based on similar applications of the riemann hypothesis over finite fields we prove general estimates with power saving for such sums we then derive various concrete applications

maximum of the characteristic polynomial of the ginibre ensemble we compute the leading asymptotics for the maximum of the centered logarithm of the absolute value of the characteristic polynomial denoted psi n of the ginibre ensemble as the dimension of the random matrix n tends to infty the method relies on the log correlated structure of the field psi n and we obtain the lower bound for the maximum by constructing a family of gaussian multiplicative chaos measures associated with certain regularization of psi n at small mesoscopic scales we also obtain the leading asymptotics for the dimensions of the sets of thick points and verify that they are consistent with the predictions coming from the gaussian free field a key technical input is the approach from ameur et al ann probab 43 3 1157 1201 2015 to derive the necessary asymptotics as well as the results from webb and wong proc lond math soc 3 118 5 1017 1056 2019

le probleme de tchebychev pour le douzieme polynome cyclotomique let p n denote the largest prime factor of the integer n and 12 n n 4 n 2 1 we prove that for x large enough we have p x x 1 c with c 10 26531

kloosterman sums for clifford algebras and a lower bound for the positive eigenvalues of the laplacian for congruence subgroups acting on hyperbolic spaces

correlations of the characteristic polynomials in the gaussian unitary ensemble or a singular hankel determinant we obtain large n asymptotics for products of powers of the absolute values of the characteristic polynomials in the gaussian unitary ensemble gue of n n matrices our results can also be interpreted as asymptotics of the determinant of a hankel matrix whose symbol is supported on the real line and possesses power like fisher hartwig singularities

asymptotics of toeplitz hankel and toeplitz hankel determinants with fisher hartwig singularities we study the asymptotics in n for n dimensional toeplitz determinants whose symbols possess fisher hartwig singularities on a smooth background we prove the general nondegenerate asymptotic behavior as conjectured by basor and tracy we also obtain asymptotics of hankel determinants on a nite interval as well as determinants of toeplitz hankel type our analysis is based on a study of the related system of orthogonal polynomials on the unit circle using the riemann hilbert approach

common solution to the lyapunov equation for 2 2 complex matrices abstract in this work we solve the problem of a common solution to the lyapunov equation for 2 2 complex matrices we show that necessary and sufficient conditions for the existence of a common solution to the lyapunov equation for 2 2 complex matrices a and b is that matrices a i i b i i and a i i 1 b i i have no negative real eigenvalues for all r we show how these results relate to a special class of 4 4 real matrices

on decomposing any matrix as a linear combination of three idempotents abstract in a recent article we gave a full characterization of matrices that can be decomposed as linear combinations of two idempotents with prescribed coefficients in this one we use those results to improve on a recent theorem of rabanovich we establish that every square matrix is a linear combination of three idempotents for an arbitrary coefficient field rather than just one of characteristic 0

on the diameters of commuting graphs the commuting graph of a ring r denoted by r is a graph whose vertices are all non central elements of r and two distinct vertices x and y are adjacent if and only if xy yx let d be a division ring and n 3 in this paper we investigate the diameters of mn d and determine the diameters of some induced subgraphs of mn d such as the induced subgraphs on the set of all non scalar non invertible nilpotent idempotent and involution matrices in mn d for every field f it is shown that if mn f is a connected graph then diam mn f 6 we conjecture that if mn f is a connected graph then diam mn f 5 we show that if f is an algebraically closed field or n is a prime number and mn f is a connected graph then diam mn f 4 finally we present some applications to the structure of pairs of idempotents which may prove of independent interest

semiclassical limits for the hyperbolic plane we study the concentration properties of high energy eigenfunctions for the laplace beltrami operator of the hyperbolic plane with special consideration of automorphic eigenfunctions at the center of our investigation is the microlocal lift of an eigenfunction to sl 2 introduced by s zelditch the microlocal lift is based on s helgason s fourier transform and has a straightforward description in terms of the lie algebra sl 2 we begin with an elementary demonstration of zelditch s exact differential equation for the microlocal lift further for a sequence of suitably bounded eigenfunctions with eigenvalues tending to infinity we show that the limit of microlocal lifts is a geodesic flow invariant positive measure our main consideration is the microlocal lifts of the elementary eigenfunctions constructed from the macdonald bessel functions we find that with a scaling of auxiliary parameters the corresponding high energy limit converges to the positive dirac measure for the lift to sl 2 of a single geodesic on the upper half plane in particular at high energy the microlocal lift of a macdonald bessel function is concentrated along the lift of a single geodesic we further find that the convergence is uniform in parameters and thus our considerations can be applied to study automorphic eigenfunctions in particular the maass cusp forms for certain cofinite subgroups of sl 2 a formula is presented for the microlocal lift of a cusp form as an integral of a family of dirac measures and a measure given directly by sums of products of the classical fourier coefficients of the cusp form the formula establishes that questions on quantum chaos for automorphic eigenfunctions are equivalent to classically stated questions on twisted sums of coefficients in particular we find that the uniform distribution for the microlocal limit is equivalent to a uniform distribution for the limit of twisted coefficient sums we extend the considerations to the case of the nonholomorphic eisenstein series for the modular group we find that a result of w luo and p sarnak is equivalent to a limit sum formula involving the riemann zeta function and the elementary divisor function the formula is suggestive of a formula of s ramanujan

characteristic polynomials of random matrices number theorists have studied extensively the connections between the distribution of zeros of the riemann function and of some generalizations with the statistics of the eigenvalues of large random matrices it is interesting to compare the average moments of these functions in an interval to their counterpart in random matrices which are the expectation values of the characteristic polynomials of the matrix it turns out that these expectation values are quite interesting for instance the moments of order 2k scale for unitary invariant ensembles as the density of eigenvalues raised to the power k 2 the prefactor turns out to be a universal number i e it is independent of the specific probability distribution an equivalent behaviour and prefactor had been found as a conjecture within number theory the moments of the characteristic determinants of random matrices are computed here as limits at coinciding points of multi point correlators of determinants these correlators are in fact universal in dyson s scaling limit in which the difference between the points goes to zero the size of the matrix goes to infinity and their product remains finite

asymptotics for the partition function in two cut random matrix models we obtain large n asymptotics for the random matrix partition function z n v int mathbb r n prod i j x i x j 2 prod j 1 ne nv x j dx j in the case where v is a polynomial such that the random matrix eigenvalues accumulate on two disjoint intervals the two cut case we compute leading and sub leading terms in the asymptotic expansion for log zn v up to terms that are small as n to infty our approach is based on the explicit computation of the first terms in the asymptotic expansion for a quartic symmetric potential v afterwards we use deformation theory of the partition function and of the associated equilibrium measure to generalize our results to general two cut potentials v the asymptotic expansion of log zn v as n to infty contains terms that depend analytically on the potential v and that have already appeared in the literature in addition our method allows us to compute the v independent terms of the asymptotic expansion of log zn v which to the best of our knowledge had not appeared before in the literature we use rigorous orthogonal polynomial and riemann hilbert techniques which had to this point only been successful to compute asymptotics for the partition function in the one cut case

the effect of assuming the identity as a generator on the length of the matrix algebra abstract let m n f be the algebra of n n matrices and let s be a generating set of m n f as an f algebra the length of a finite generating set s of m n f is the smallest number k such that words of length not greater than k generate m n f as a vector space traditionally the identity matrix is assumed to be automatically included in all generating sets s and counted as a word of length 0 in this paper we discuss how the problem changes if this assumption is removed

rankin selberg l functions in the level aspect keywords moments rankin selberg convolution level aspect convexity breaking reference tan article 2002 003doi 10 1215 s0012 7094 02 11416 1 record created on 2008 11 14 modified on 2017 05 12

on the asymptotics of some large hankel determinants generated by fisher hartwig symbols defined on the real line we investigate the asymptotics of hankel determinants of the form detj k 0n 1 dx n x i 1m i x 2qixj k as n with q and fixed where is an infinite subinterval of r and n x is a positive weight on such objects are natural analogs of toeplitz determinants generated by fisher hartwig symbols and arise in random matrix theory in the investigation of certain expectations involving random characteristic polynomials the reduced density matrices of certain one dimensional systems of trapped impenetrable bosons can also be expressed in terms of hankel determinants of this form we focus on the specific cases of scaled hermite and laguerre weights we compute the asymptotics by using a duality formula expressing the n n hankel determinant as a 2 q1 qm fold integral which is valid when each qi is natural we thus verify for such q a recent conjecture of forrester and frankel derived using a log gas argument

bilinear forms with kloosterman sums and applications we prove nontrivial bounds for general bilinear forms in hyper kloosterman sums when the sizes of both variables may be below the range controlled by fourier analytic methods polya vinogradov range we then derive applications to the second moment of cusp forms twisted by characters modulo primes and to the distribution in arithmetic progressions to large moduli of certain eisenstein hecke coefficients on gl 3 our main tools are new bounds for certain complete sums in three variables over finite fields proved using methods from algebraic geometry especially l adic cohomology and the riemann hypothesis

universal results for correlations of characteristic polynomials riemann hilbert approach we prove that general correlation functions of both ratios and products of characteristic polynomials of hermitian random matrices are governed by integrable kernels of three different types a those constructed from orthogonal polynomials b those constructed from cauchy transforms of the same orthogonal polynomials and finally c those constructed from both orthogonal polynomials and their cauchy transforms these kernels are related with the riemann hilbert problem for orthogonal polynomials for the correlation functions we obtain exact expressions in the form of determinants of these kernels derived representations enable us to study asymptotics of correlation functions of characteristic polynomials via the deift zhou steepest descent stationary phase method for riemann hilbert problems and in particular to find negative moments of characteristic polynomials this reveals the universal parts of the correlation functions and moments of characteristic polynomials for an arbitrary invariant ensemble of 2 symmetry class

fractional brownian motion with hurst index h 0 and the gaussian unitary ensemble the goal of this paper is to establish a relation between characteristic polynomials of n n gue random matrices h as n and gaussian processes with logarithmic correlations we introduce a regularized version of fractional brownian motion with zero hurst index which is a gaussian process with stationary increments and logarithmic increment structure then we prove that this process appears as a limit of d n z log det h zi on mesoscopic scales as n by employing a fourier integral representation we use this to prove a continuous analogue of a result by diaconis and shahshahani j appl probab 31a 1994 49 62 on the macroscopic scale d n x gives rise to yet another type of gaussian process with logarithmic correlations we give an explicit construction of the latter in terms of a chebyshev fourier random series

on the kuznetsov trace formula for pgl2 c abstract in this note using a representation theoretic method of cogdell and piatetski shapiro we prove the kuznetsov trace formula for an arbitrary discrete group in pgl 2 c that is cofinite but not cocompact an essential ingredient is a kernel formula recently proved by the author on bessel functions for pgl 2 c this approach avoids the difficult analysis in the existing method due to bruggeman and motohashi

a central limit theorem for the determinant of a wigner matrix abstract we establish a central limit theorem for the log determinant log det m n of a wigner matrix m n under the assumption of four matching moments with either the gue or goe ensemble more specifically we show that this log determinant is asymptotically distributed like n log n 1 2 log n 1 2 log n r when one matches moments with gue and n log n 1 4 log n 1 4 log n r when one matches moments with goe

autour de la conjecture de sato tate pour les sommes de kloosterman i keywords almost primes sieve identity of buchstab type kloosterman sum sato tate conjecture distribution function sato tate distribution estimates for multiple exponential sums reference tan article 1995 001doi 10 1007 bf01884290 record created on 2008 11 14 modified on 2017 05 12

nonintersecting brownian motions on the half line and discrete gaussian orthogonal polynomials we study the distribution of the maximal height of the outermost path in the model of n nonintersecting brownian motions on the half line as n showing that it converges in the proper scaling to the tracy widom distribution for the largest eigenvalue of the gaussian orthogonal ensemble this is as expected from the viewpoint that the maximal height of the outermost path converges to the maximum of the airy2 process minus a parabola our proof is based on riemann hilbert analysis of a system of discrete orthogonal polynomials with a gaussian weight in the double scaling limit as this system approaches saturation we consequently compute the asymptotics of the free energy and the reproducing kernel of the corresponding discrete orthogonal polynomial ensemble in the critical scaling in which the density of particles approaches saturation both of these results can be viewed as dual to the case in which the mean density of eigenvalues in a random matrix model is vanishing at one point

l adic gkz hypergeometric sheaves and exponential sums abstract to a torus action on a complex vector space gelfand kapranov and zelevinsky introduce a system of differential equations called the gkz hypergeometric system its solutions are gkz hypergeometric functions we study the l adic counterpart of the gkz hypergeometric system which we call the l adic gkz hypergeometric sheaf it is an object in the derived category of l adic sheaves on the affine space over a finite field traces of frobenius on stalks of this object at rational points of the affine space define the hypergeometric functions over the finite field introduced by gelfand and graev we prove that the l adic gkz hypergeometric sheaf is perverse calculate its rank and prove that it is irreducible under the non resonance condition we also study the weight filtration of the gkz hypergeometric sheaf determine its lisse locus and apply our result to the study of weights of twisted exponential sums

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

universality for eigenvalue correlations at the origin of the spectrum we establish universality of local eigenvalue correlations in unitary random matrix ensembles frac 1 z n det m 2 alpha e n rm tr v m dm near the origin of the spectrum if v is even and if the recurrence coefficients of the orthogonal polynomials associated with x 2 e nv x have a regular limiting behavior then it is known from work of akemann et al and kanzieper and freilikher that the local eigenvalue correlations have universal behavior described in terms of bessel functions we extend this to a much wider class of confining potentials v our approach is based on the steepest descent method of deift and zhou for the asymptotic analysis of riemann hilbert problems this method was used by deift et al to establish universality in the bulk of the spectrum a main part of the present work is devoted to the analysis of a local riemann hilbert problem near the origin

bilinear forms in weyl sums for modular square roots and applications abstract let q be a prime p 1 and let n q p denote the number of rational primes p p that split in the imaginary quadratic field q q the first part of this paper establishes various unconditional and conditional under existence of a siegel zero lower bounds for n q p in the range q 1 4 e p q for any fixed e 0 this improves upon what is implied by work of pollack and benli pollack the second part of this paper is dedicated to proving an estimate for a bilinear form involving weyl sums for modular square roots equivalently salie sums our estimate has a power saving in the so called polya vinogradov range and our methods involve studying an additive energy coming from quadratic residues in f q this bilinear form is inspired by the recent automorphic motivation the second moment for twisted l functions attached to kohnen newforms has recently been computed by the first and fourth authors so the third part of this paper links the above two directions together and outlines the arithmetic applications of this bilinear form these include the equidistribution of quadratic roots of primes products of primes and relaxations of a conjecture of erd s odlyzko sarkozy

universal correlations of trapped one dimensional impenetrable bosons we calculate the asymptotic behaviour of the one body density matrix of one dimensional impenetrable bosons in finite size geometries our approach is based on a modification of the replica method from the theory of disordered systems we obtain explicit expressions for oscillating terms similar to fermionic friedel oscillations these terms are universal and originate from the strong short range correlations between bosons in one dimension

toeplitz determinants with merging singularities we study asymptotic behavior for determinants of n n toeplitz matrices corresponding to symbols with two fisher hartwig singularities at the distance 2t 0 from each other on the unit circle we obtain large n asymptotics which are uniform for 0 t t0 where t0 is fixed they describe the transition as t 0 between the asymptotic regimes of 2 singularities and 1 singularity the asymptotics involve a particular solution to the painlev e v equation we obtain small and large argument expansions of this solution as applications of our results we prove a conjecture of dyson on the largest occupation number in the ground state of a one dimensional bose gas and a conjecture of fyodorov and keating on the second moment of powers of the characteristic polynomials of random matrices

the fourth moment of dirichlet l functions we compute the fourth moment of dirichlet l functions at the central point for prime moduli with a power savings in the error term

real zeros and size of rankin selberg l functions in the level aspect in this paper some asymptotic formulas are proved for the harmonic mollified second moment of a family of rankin selberg l functions one of the main new input is a substantial improvement of the admissible length of the mollifier which is done by solving a shifted convolution problem by a spectral method on average a first consequence is a new subconvexity bound for rankin selberg l functions in the level aspect moreover infinitely many rankin selberg l functions having at most eight non trivial real zeros are produced and some new non trivial estimates for the analytic rank of the family studied are obtained

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

density results for automorphic forms on hilbert modular groups ii we obtain an asymptotic formula for a weighted sum over cuspidal eigenvalues in a specific region for sl2 over a totally real number field f with a discrete subgroup of hecke type 0 i for a non zero ideal i in the ring of integers of f the weights are products of fourier coefficients this implies in particular the existence of infinitely many cuspidal automorphic representations with multi eigenvalues in various regions growing to infinity for instance in the quadratic case the regions include floating boxes floating balls sectors slanted strips see 1 2 4 1 2 13 and products of prescribed small intervals for all but one of the infinite places of f the main tool in the derivation is a sum formula of kuznetsov type sum formula for sl2 over a totally real number field theorem 2 1