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

on the minimum ropelength of knots and links the ropelength of a knot is the quotient of its length by its thickness the radius of the largest embedded normal tube around the knot we prove existence and regularity for ropelength minimizers in any knot or link type these are c 1 1 curves but need not be smoother we improve the lower bound for the ropelength of a nontrivial knot and establish new ropelength bounds for small knots and links including some which are sharp

entropy of quantum limits in this paper we show that any measure arising as a weak limit of microlocal lifts of eigenfunctions of the laplacian on certain arithmetic manifolds have dimension at least 11 9 and in particular all ergodic components of this measure with respect to the geodesic flow have positive entropy

recent progress on the quantum unique ergodicity conjecture we report on some recent striking advances on the quantum unique ergodicity or que conjecture concerning the distribution of large frequency eigenfunctions of the laplacian on a negatively curved manifold the account falls naturally into two categories the first concerns the general conjecture where the tools are more or less limited to micro local analysis and the dynamics of the geodesic flow the second is concerned with arithmetic such manifolds where tools from number theory and ergodic theory of flows on homogeneous spaces can be combined with the general methods to resolve the basic conjecture as well as its holomorphic analogue the recent account by zelditch zel4 covers some of the same material and his and our discussion complement each other nicely as he goes into more detail with the first category and we with the second this note is not meant to be a survey of these topics and the discussion is not chronological our aim is to expose these recent developments after introducing the necessary backround which places them in their proper context recent progress on que 2 1 the general que conjecture in figure 1 the domains e s and b are an ellipse a stadium and a barnett billiard table respectively superimposed on these are the densities of a consecutive sequence of high frequency eigenfunctions states modes of the laplacian that is they are solutions to 4 j j j 0 in 0 dirichlet boundary conditions j dxdy 1 0 here 4 divgrad 2 x2 2 y2 1 2 3 are the eigenvalues and the eigenfunctions are normalized to have unit l norm the sequences are of 12 consecutive modes around the 5600 eigenvalue they are ordered from left to right and then down and the grayscale represents the probability density with zero white and larger values darker

on the number of nodal domains of random spherical harmonics let n f be a number of nodal domains of a random gaussian spherical harmonic f of degree n we prove that as n grows to infinity the mean of n f n 2 tends to a positive constant a and that n f n 2 exponentially concentrates around a this result is consistent with predictions made by bogomolny and schmit using a percolation like model for nodal domains of random gaussian plane waves

l infty norms of eigenfunctions of arithmetic surfaces here and elsewhere a l1o0 as a xc is a basic problem of quantum chaos s in any case almost nothing beyond 0 2 is known about 110 joc when the curvature is negative one can push the standard techniques and replace a1 4 by a1 4 log a in this case in this paper we use arithmetic techniques in particular modular correspondences to obtain the first improvement in the exponent 0 2 we also obtain lower bounds on the l norms in these arithmetic cases in more detail let a b be a quaternion division algebra over q a is linearly generated by 1 w q wq over q and w2 a q2 b wq qw 0 here a b e z are square free and we will assume that a 0 as usual the norm and trace are defined by n a a d tr a a oy where if a xo x1w x2q x3wq

joint quasimodes positive entropy and quantum unique ergodicity we study joint quasimodes of the laplacian and one hecke operator on compact congruence surfaces and give conditions on the orders of the quasimodes that guarantee positive entropy on almost every ergodic component of the corresponding semiclassical measures together with the measure classification result of lindenstrauss ann math 2 163 1 165 219 2006 this implies quantum unique ergodicity for such functions our result is optimal with respect to the dimension of the space from which the quasi mode is constructed we also study equidistribution for sequences of joint quasimodes of the two partial laplacians on compact irreducible quotients of mathbb h times mathbb h

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

invariant measures and arithmetic quantum unique ergodicity we classify measures on the locally homogeneous space i sl 2 r l which are invariant and have positive entropy under the diagonal subgroup of sl 2 r and recurrent under l this classification can be used to show arithmetic quantum unique ergodicity for compact arithmetic surfaces and a similar but slightly weaker result for the finite volume case other applications are also presented in the appendix joint with d rudolph we present a maximal ergodic theorem related to a theorem of hurewicz which is used in theproof of the main result

arithmetic and equidistribution of measures on the sphere motivated by problems of mathematical physics quantum chaos questions of equidistribution of eigenfunctions of the laplace operator on a riemannian manifold have been studied by several authors we consider here in analogy with arithmetic hyperbolic surfaces orthonormal bases of eigenfunctions of the laplace operator on the two dimensional unit sphere which are also eigenfunctions of an algebra of hecke operators which act on these spherical harmonics we formulate an analogue of the equidistribution of mass conjecture for these eigenfunctions as well as of the conjecture that their moments tend to moments of the gaussian as the eigenvalue increases for such orthonormal bases we show that these conjectures are related to the analytic properties of degree eight arithmetic l functions associated to triples of eigenfunctions moreover we establish the conjecture for the third moments and give a conditional on standard analytic conjectures about these arithmetic l functions proof of the equidistribution of mass conjecture

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

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

nodal domains statistics a criterion for quantum chaos we consider the distribution of the properly normalized numbers of nodal domains of wave functions in 2d quantum billiards we show that these distributions distinguish clearly between systems with integrable separable or chaotic underlying classical dynamics and for each case the limiting distribution is universal system independent thus a new criterion for quantum chaos is provided by the statistics of the wave functions which complements the well established criterion based on spectral statistics

antimagic labelings of caterpillars abstract a k antimagic labeling of a graph g is an injection from e g to 1 2 e g k such that all vertex sums are pairwise distinct where the vertex sum at vertex u is the sum of the labels assigned to edges incident to u we call a graph k antimagic when it has a k antimagic labeling and antimagic when it is 0 antimagic hartsfield and ringel conjectured that every simple connected graph other than k2 is antimagic but the conjecture is still open even for trees here we study k antimagic labelings of caterpillars we use algorithmic and constructive techniques instead of the standard combinatorial nullstellensatz method to prove our results i any caterpillar of order n is n 1 2 2 antimagic ii any caterpillar with a spine of order s with either at least 3 s 1 2 leaves or s 1 2 consecutive vertices of degree at most 2 at one end of a longest path is antimagic and iii if p is a prime number any caterpillar with a spine of order p p 1 or p 2 is 1 antimagic

the antimagicness of the cartesian product of graphs an antimagic labeling of a graph with m edges and n vertices is a bijection from the set of edges to the set 1 2 3 m such that all the nvertex sums are pairwise distinct where the vertex sum of a vertex v is the sum of labels of all edges incident with v a graph is called antimagic if it has an antimagic labeling the antimagicness of the cartesian product of graphs in several special cases has been studied tao ming wang toroidal grids are anti magic in proc 11th annual international computing and combinatorics conference coocoon 2005 in lncs vol 3595 springer 2005 pp 671 679 yongxi cheng a new class of antimagic cartesian product graphs discrete mathematics 308 24 2008 6441 6448 in this paper we develop new construction methods that are applied to more general cases we prove that the cartesian product of paths is antimagic if one of them has at least three edges this almost answers the open problems in yongxi cheng lattice grids and prisms are antimagic theoretical computer science 374 2007 66 73 we also prove that the cartesian product of an antimagic regular graph and a connected graph is antimagic which extends the results of the latter of the two references where several special cases are studied

numerical upper bounds on ropelengths of large physical knots numerical computations explored for this paper show that an upper bound on the ropelength of large knots with crossing number n grows as fast as nln2n the algorithms to randomly generate samples of such large knots and to determine an upper bound on the ropelength for each knot are described the numeric results are presented and compared to the smallest known theoretical upper bounds on ropelength

nodal intersections for random eigenfunctions on the torus we investigate the number of nodal intersections of random gaussian laplace eigenfunctions on the standard two dimensional flat torus arithmetic random waves with a fixed smooth reference curve with nonvanishing curvature the expected intersection number is universally proportional to the length of the reference curve times the wavenumber independent of the geometry our main result prescribes the asymptotic behavior of the nodal intersections variance for smooth curves in the high energy limit remarkably it is dependent on both the angular distribution of lattice points lying on the circle with radius corresponding to the given wavenumber and the geometry of the given curve in particular this implies that the nodal intersection number admits a universal asymptotic law with arbitrarily high probability

knots in self avoiding walks discusses the existence of knots in random self avoiding walks on a lattice using kesten s 1963 pattern theorem it is shown that almost all sufficiently long self avoiding walks on the three dimensional simple cubic lattice contain a knot

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

entropy and the localization of eigenfunctions we study the large eigenvalue limit for the eigenfunctions of the laplacian on a compact manifold of negative curvature in fact we only assume that the geodesic flow has the anosov property in the semi classical limit we prove that the wigner measures associated to eigenfunctions have positive metric entropy in particular they cannot concentrate entirely on closed geodesies 1 introduction statement of results we consider a compact riemannian manifold m of dimension d 2 and assume that the geodesic flow ter acting on the unit tangent bundle of m has a chaotic behaviour this refers to the asymptotic properties of the flow when time t tends to infinity ergodicity mixing hyperbolicity we assume here that the geodesic flow has the anosov property the main example being the case of negatively curved manifolds the words quantum chaos express the intuitive idea that the chaotic features of the geodesic flow should imply certain special features for the corresponding quantum dynamical system that is according to schrodinger the unitary flow exp iht t r acting on the hilbert space l2 m where a stands for the laplacian on m and h is proportional to the planck constant recall that the quantum flow converges in a sense to the classical flow g in the so called semi classical limit h 0 one can imagine that for small values ofh the quantum system will inherit certain qualitative properties of the classical flow one expects for instance a very different behaviour of eigenfunctions of the laplacian or the distribution of its eigenvalues if the geodesic flow is anosov or in the other extreme completely integrable see sa95 the convergence of the quantum flow to the classical flow is stated in the egorov theorem consider one of the usual quantization procedures op which associates an operator oph a acting on l2 m to every smooth compactly supported function a e c t m on the cotangent bundle t m according to the egorov theorem we have for any fixed t

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

lattice grids and prisms are antimagic an antimagic labelling of a finite undirected simple graph with m edges and n vertices is a bijection from the set of edges to the integers 1 m such that all n vertex sums are pairwise distinct where a vertex sum is the sum of labels of all edges incident with the same vertex a graph is called antimagic if it has an antimagic labelling in 1990 hartsfield and ringel conjectured that every connected graph but k 2 is antimagic in t m wang toroidal grids are antimagic in proc 11th annual international computing and combinatorics conference cocoon 2005 in lncs vol 3595 springer 2005 pp 671 679 wang showed that the toroidal grids the cartesian products of two or more cycles are antimagic two open problems left in wang s paper are about the antimagicness of lattice grid graphs and prism graphs which are the cartesian products of two paths and of a cycle and a path respectively in this article we prove that these two classes of graphs are antimagic by constructing such antimagic labellings

the modulus of continuity for γ0 m semi classical limits we study the behavior of a large eigenvalue limit of eigenfunctions for the hyperbolic laplacian for the modular quotient sl 2 fejer summation and results of s zelditch are used to show that the microlocal lifts of eigenfunctions have large eigenvalue limit a geodesic flow invariant measure for the modular unit cotangent bundle the limit is studied for hecke maass forms joint eigenfunctions of the hecke operators and the hyperbolic laplacian the first modulus of continuity result is presented for the limit the singular concentration set of the limit cannot be a compact union of closed geodesics and measured geodesic laminations

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

how many zeros of a random polynomial are real we provide an elementary geometric derivation of the kac integral formula for the expected number of real zeros of a random polynomial with independent standard normally distributed coefficients we show that the expected number of real zeros is simply the length of the moment curve 1 t tn projected onto the surface of the unit sphere divided by the probability density of the real zeros is proportional to how fast this curve is traced out we then relax kac s assumptions by considering a variety of random sums series and distributions and we also illustrate such ideas as integral geometry and the fubini study metric

uniform distribution of eigenfunctions on compact hyperbolic surfaces on demontre que les fonctions propres k du laplacien sur une surface hyperbolique compacte x devient uniformement distribuee sur x quand k

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

the mean square of the error term for the fourth power moment of the zeta function we prove that j e2 t 2 dt t 2 log c t where e2 t denotes the error term in the asymptotic formula for fo k of dt the proof depends on the second author s explicit formula for a modified version of the latter integral several related results are also proved including the analogue of the above result for the binary additive divisor problem

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

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

a geometric characterization of arithmetic fuchsian groups the trace set of a fuchsian group encodes the set of lengths of closed geodesics in the surface h luo and sarnak showed that the trace set of a cofinite arithmetic fuchsian group satisfies the bounded clustering property sarnak then conjectured that the b c property actually characterizes arithmetic fuchsian groups schmutz stated the even stronger conjecture that a cofinite fuchsian group is arithmetic if its trace set has linear growth he proposed a proof of this conjecture in the case when the group contains at least one parabolic element but unfortunately this proof contains a gap in the present paper we point out this gap and we prove sarnak s conjecture under the assumption that the fuchsian group contains parabolic elements

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

localization in infinite billiards a comparison between quantum and classical ergodicity consider the non compact billiard in the first quandrant bounded by the positive x semiaxis the positive y semiaxis and the graph of f x x 1 1 2 although the schnirelman theorem holds the quantum average of the position xis finite on any eigenstate while classical ergodicity entails that the classical time average of xis unbounded

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

the behaviour of eigenstates of arithmetic hyperbolic manifolds in this paper we study some problems arising from the theory of quantum chaos in the context of arithmetic hyperbolic manifolds we show that there is no strong localization scarring onto totally geodesic submanifolds arithmetic examples are given which show that the random wave model for eigenstates does not apply universally in 3 degrees of freedom