on the characteristic polynomial of a random unitary matrix we present a range of fluctuation and large deviations results for the logarithm of the characteristic polynomial z of a random n n unitary matrix as n first we show that evaluated at a finite set of distinct points is asymptotically a collection of i i d complex normal random variables this leads to a refinement of a recent central limit theorem due to keating and snaith and also explains the covariance structure of the eigenvalue counting function next we obtain a central limit theorem for ln z in a sobolev space of generalised functions on the unit circle in this limiting regime lower order terms which reflect the global covariance structure are no longer negligible and feature in the covariance structure of the limiting gaussian measure large deviations results for ln z a evaluated at a finite set of distinct points can be obtained for for higher order scalings we obtain large deviations results for ln z a evaluated at a single point there is a phase transition at a ln n which only applies to negative deviations of the real part reflecting a switch from global to local conspiracy

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

zeros of dedekind zeta functions in the critical strip in this paper we describe a computation which established the grh to height 92 resp 40 for cubic number fields resp quartic number fields with small discriminant we use a method due to e friedman for computing values of dedekind zeta functions we take care of accumulated roundoff error to obtain results which are mathematically rigorous and we generalize turing s criterion to prove that there is no zero off the critical line we finally give results concerning the grh for cubic and quartic fields tables of low zeros for number fields of degree 5 and 6 and statistics about the smallest zero of a number field

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

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

bounds for twisted symmetric square l functions iii abstract let f be a hecke modular form and let be a primitive character of conductor q l assume that q is an odd prime in this paper we prove the subconvex bound l 1 2 sym 2 f f q e q 3 l 1 4 1 36 e for any e 0 this can be compared with the recently established t aspect subconvexity of the symmetric square l functions

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

on central l derivative values of automorphic forms we study the average and nonvanishing of the central l derivative values of l s f and l s f k scriptscriptstyle d for f in an orthogonal hecke eigenbasis mathcal h 2k of weight 2k cusp forms of level 1 for large odd k here f k scriptscriptstyle d is the base change of f to an imaginary quadratic field k scriptscriptstyle d mathbb q sqrt d with fundamental discriminant d we prove asymptotic formulas for the first and second moments of l frac 1 2 f as well as the first moment of l frac 1 2 f k scriptscriptstyle d over mathcal h 2k as odd k rightarrow infty further we employ mollifiers to establish that for sufficiently large k there are positive proportion of hecke eigenforms f in mathcal h 2k with l frac 1 2 f ne 0 we also give applications of our results to heegner cycles of high weights of the modular curve x 0 1

on the central value of symmetric square l functions let s k n be the space of cusp forms of weight k level n and character for f in s k n chi let l s sym2 f be the symmetric square l function and l s f otimes f be the rankin selberg square attached to f for fixed k 2 n prime and real primitive asymptotic formulas for the first and second moment of the central value of l s sym2 f and l s f otimes f over a basis of s k n are given as n as an application it is shown that a positive proportion of the central values l 1 2 sym2 f does not vanish

composition is almost but not quite as good as s 1 1 we establish a polynomial upper bound on the time complexity of an s 1 1 function in programming system with a linear time composition function this improves the doubly exponential upper bound of machtey and young 1978 the only previously known upper bound and invalidates the belief expressed twice in the literature machtey 1978 machtey and young 1978 that it could not be significantly improved we then show our upper bound to be tight by exhibiting a family of acceptable programming systems for which it is optimal we deduce several bounds on the time complexity of composition functions s 1 1 functions and various other semantic transformations of programs in programming systems with a linear or polynomial time composition function in particular we show the existence of an acceptable programming system with a quadratic time composition function but no subexponential time s 1 1 function in one interpretation from marcoux 1991 this last result states that the complexity of a composition function for an effective programming system does not give an upper bound on the complexity of the task of programming in that programming system by contrast results by royer 1987 indicate that this task is essentially no more complex than computing an s 1 1 function

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

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

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

explicit bounds for primes in residue classes let e k be an abelian extension of number fields with e q let and n denote the absolute discriminant and degree of e let denote an element of the galois group of e k we prove the following theorems assuming the extended riemann hypothesis 1 there is a degree 1 prime p of k such that p e k satisfying np 1 o 1 log 2n 2 2 there is a degree 1 prime p of k such that p e k generates the same group as satisfying np 1 o 1 log 2 3 for k q there is a prime p such that p e q satisfying p 1 o 1 log 2 in 1 and 2 we can in fact take p to be unramified in k q a special case of this result is the following 4 if gcd m q 1 the least prime p m mod q satisfies p 1 o 1 q log q 2 it follows from our proof that 1 3 also hold for arbitrary galois extensions provided we replace by its conjugacy class our theorems lead to explicit versions of 1 4 including the following the least prime p m mod q is less than 2 q log q 2

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

exponential decay in the frequency of analytic ranks of automorphic l functions this note should be seen as a appendum to the work of e kowalski and the second author km1 and deals with the problem of bounding unconditionnally the order of vanishing at the critical point in a family of l functions this problem is illustrated in the particular case of l functions of weight 2 primitive modular forms of prime level recall the notations from km1 for q a prime number let s2 q be the set of primitive forms of weight 2 and level q normalized so that their first fourier coefficient is 1 for

a note on natural complete sets and godel numberings creative sets or the complete recursively enumerable sets play an important role in logic and mathematics and they are known to be recursively isomorphic therefore on the one hand all the complete sets can be viewed as equivalent on the other hand we intuitively perceive some complete sets as more natural and simpler than others in this note we try to capture this intuitive concept precisely by defining a complete set to be natural if all other recursively enumerable sets can be reduced to it by computationally simple reductions and show that these natural complete sets are all isomorphic under the same type of computationally simple mappings the same ideas are applied to define natural godel numberings and using natural godel numberings to show that natural creative sets coincide with the natural complete sets

an explicit trace formula of jacquet zagier type for hilbert modular forms abstract we give an exact formula of the average of adjoint l functions of holomorphic hilbert cusp forms with a fixed weight and a square free level which is a generalization of zagier s formula known for the case of elliptic cusp forms on sl 2 z as an application we prove that the satake parameters of hilbert cusp forms with a fixed weight and with growing square free levels are equidistributed in an ensemble constructed by values of the adjoint l functions

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

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

nonvanishing of central l values of maass forms abstract with the method of moments and the mollification method we study the central l values of gl 2 maass forms of weight 0 and level 1 and establish a positive proportional nonvanishing result of such values in the aspect of large spectral parameter in short intervals which is qualitatively optimal in view of weyl s law as an application of this result and a formula of katok sarnak we give a nonvanishing result on the first fourier coefficients of maass forms of weight 1 2 and level 4 in the kohnen plus space

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

weyl group multiple dirichlet series constructed from quadratic characters we construct multiple dirichlet series in several complex variables whose coefficients involve quadratic residue symbols the series are shown to have an analytic continuation and satisfy a certain group of functional equations these are the first examples of an infinite collection of unstable weyl group multiple dirichlet series in greater than two variables having the properties predicted in 2

bertrand s postulate for primes in arithmetical progressions abstract bertrand s postulate is the theorem that the interval x 2 x contains at least one prime for x 2 1 we prove building on work of erd os analogues of this result in which the interval is of the form x zx and there are at least m primes a mod d required to be contained in this interval and where z a and d have to satisfy some conditions for the case m 1 the results are worked out using a computer they can be found in table 1

computing l functions with large conductor an algorithm is given to efficiently compute l functions with large conductor in a restricted range of the critical strip examples are included for about 24000 dihedral galois representations with conductor near 10 7 the data shows good agreement with a symplectic random matrix model

autocorrelation of random matrix polynomials we calculate the autocorrelation functions or shifted moments of the characteristic polynomials of matrices drawn uniformly with respect to haar measure from the groups u n o 2n and usp 2n in each case the result can be expressed in three equivalent forms as a determinant sum and hence in terms of symmetric polynomials as a combinatorial sum and as a multiple contour integral these formulae are analogous to those previously obtained for the gaussian ensembles of random matrix theory but in this case are identities for any size of matrix rather than large matrix asymptotic approximations they also mirror exactly the autocorrelation formulae conjectured to hold for l functions in a companion paper this then provides further evidence in support of the connection between random matrix theory and the theory of l functions

exact controllability and boundary stabilization of flexural vibrations of an internally damped flexible space structure we consider a problem of flexural vibrations of a flexible rectangular panel incorporating small internal damping of the material the panel is hoisted by a rigid hub at one end and the other end totally free for attachment of the rigid hub on one side of the panel its dynamics leads to a nonstandard boundary condition and the whole system becomes a hybrid system of dynamics the exact controllability theory is established using the hilbert uniqueness method by means of an active control applied only on the rigid hub the uniform exponential decay of solution is also obtained directly for such formulation with viscous boundary damping at the hub end only

on the moments of the moments of the characteristic polynomials of random unitary matrices denoting by p n a theta det i ae i theta the characteristic polynomial on the unit circle in the complex plane of an n times n random unitary matrix a we calculate the kth moment defined with respect to an average over a in u n of the random variable corresponding to the 2 beta th moment of p n a theta with respect to the uniform measure frac d theta 2 pi for all k beta in mathbb n these moments of moments have played an important role in recent investigations of the extreme value statistics of characteristic polynomials and their connections with log correlated gaussian fields our approach is based on a new combinatorial representation of the moments using the theory of symmetric functions and an analysis of a second representation in terms of multiple contour integrals our main result is that the moments of moments are polynomials in n of degree k 2 beta 2 k 1 this resolves a conjecture of fyodorov and keating philos trans r soc a 372 2007 20120503 2014 concerning the scaling of the moments with n as n rightarrow infty for k beta in mathbb n indeed it goes further in that we give a method for computing these polynomials explicitly and obtain a general formula for the leading coefficient

mollification of the fourth moment of automorphic l functions and arithmetic applications we compute the asymptotics of twisted fourth power moments of modular l functions of large prime level near the critical line this allows us to prove some new non vanishing results on the central values of automorphic l functions in particular those obtained by base change from gl 2 q to gl 2 k for k a cyclic field of low degree

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

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

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

boundary stabilization for a hybrid system of viscoelasticity in this paper we study the boundary feedback stabilization of a hybrid system that is composed of a viscoelastic thin plate with one part of its edge clamped and the remaining free part attached to a viscoelastic rigid body by adopting the frequency domain method we prove that the boundary feedback controls together with the dissipation induced by the memory effect are strong enough to secure the result of the exponential stability of energy we also reach the result of the exponential stability of the hybrid system on the domain with corners without any extra hypotheses on the regularity of solutions

mock gaussian behaviour for linear statistics of classical compact groups we consider the scaling limit of linear statistics for eigenphases of a matrix taken from one of the classical compact groups we compute their moments and find that the first few moments are gaussian whereas the limiting distribution is not the precise number of gaussian moments depends upon the particular statistic considered

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 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

on the non vanishing of elliptic curve l functions at the central point we show that a large number of elliptic curve l functions do not vanish at the central point conditionally on the generalized riemann hypothesis and on an assumption on the regular distribution of the root number

control structures in hypothesis spaces the influence on learning in any learnability setting hypotheses are conjectured from some hypothesis space studied herein are the influence on learnability of the presence or absence of certain control structures in the hypothesis space first presented are control structure characterizations of some rather specific but illustrative learnability results the presence of these control structures is thereby shown essential to maintain full learning power then presented are the main theorems each of these non trivially characterizes the invariance of a learning class over hypothesis space v and the presence of a particular projection control structure called proj in v as v has suitable instances of all denotational control structures in a sense then proj epitomizes the control structures whose presence need not help and whose absence need not hinder learning power

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

the prime number race and zeros of l functions off the critical line we examine the effects of certain hypothetical configurations of zeros of dirichlet l functions lying off the critical line on the distribution of primes in arithmetic progressions

difference system for selberg correlation integrals the selberg correlation integrals are averages of the products with respect to the selberg density our interest is in the case m 1 1 when this corresponds to the th moment of the corresponding characteristic polynomial we give the explicit form of an n 1 n 1 matrix linear difference system in the variable which determines the average and we give the gauss decomposition of the corresponding n 1 n 1 matrix for a positive integer the difference system can be used to efficiently compute the power series defined by this average

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 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

dynamical boundary control for elastic plates of general shape the control of transverse vibrations of elastic plates of general shape by feedback boundary control is formulated as an abstract evolution equation because the control acts locally on the boundary which possesses a flanged rim with inertial properties of mass and bending moment the analysis concerns dynamical controllability and stabilizability of a hybrid system by the approach of energy decay inequalities and hormander s global uniqueness theorem it is shown that the system is strongly stabilizable by a locally supported damping feedback of boundary velocity and boundary angular velocity and hence the system is approximately controllable

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

explicit estimates for the error term in the prime number theorem for arithmetic progressions we give explicit numerical estimates for the chebyshev functions 4 x k l and g x k l for certain nonexceptional moduli k for values of e and b a constant c is tabulated such that 14 x k l x 9p k l exp c log2 k and k 10b the methods are similar to those used by rosser and schoenfeld in the case k 1 but are based on explicit estimates of n t x and an explicit zero free region for dirichlet l functions

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

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

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

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