simple zeros of the zeta function of a quadratic number field i let k be a fixed quadratic extension of q and write k s for the dedekind zeta function of k where s it it is wellknown and easy to prove that the number nk t of zeros of k s in the region 0 1 0 t t satisfies text n text k left text t right frac text t pi log t as t on the other hand not much is known about the number of text n text k text left text t right that are simple

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

triangles and efficiency factors summary we show that the number of triangles in the variety concurrence graph of a regulargraph design can be used to derive upper bounds on the harmonic mean efficiency factor the best alpha lattice designs and the best cyclic designs have efficiency factors very close to the bounds our investigation gives insight into the structure of such designs we see also that a certain type of regular graph group divisible design has a minimal number of triangles

cuspidality of symmetric powers with applications the purpose of this paper is to prove that the symmetric fourth power of a cusp form on rm gl 2 whose existence was proved earlier by the first author is cuspidal unless the corresponding automorphic representation is of dihedral tetrahedral or octahedral type as a consequence we prove a number of results toward the ramanujan petersson and sato tate conjectures in particular we establish the bound q sp 1 9 sb v for unramified hecke eigenvalues of cusp forms on rm gl 2 over an arbitrary number field this is the best bound available at present

moment inequalities via optimal design theory abstract a unified approach is used to rediscover a class of moment inequalities in particular complementary inequalities in respect to ratios and differences of means and also inequalities complementary to both holder s and minkowski s inequalities each of which have been established in the literature by ad hoc methods are rediscovered the impetus of the paper is that this unified method is simple and illuminating a class of optimization problems is considered in which optimization is with respect to a probability distribution moment inequalities are implied by the solutions equality occuring under the optimizing distribution however we do not claim to prove any fundamentally new inequalities at this juncture the method used is illuminating in that it first discovers the support points of the optimizing distribution it then proves to be simple to discover the optimizing probabilities the method employs the differential calculus approach developed to establish optimality conditions for optimal linear regression designs however the link with optimal design theory is not just this tenous one results in respect of optimal design criteria of themselves have implications concerning moments including the multivariate generalization of the nonnegativity of a variance

effect of particle size distribution and dynamics on the performance of two dimensional packing abstract extensive computer simulation is used to revisit and to generalize two classical problems i the random car parking dynamics of a renyi and ii the irreversible random sequential adsorption rsa of parallel squares of same size on a planar substrate of area l 2 in this paper differently from the classical rsa the squares obey the size distribution n a n 1 a where a 1 2 3 is the area of the squares using this scaling distribution and three classes of packing dynamics we study the final packing fraction of particles l and in particular its thermodynamic limit l we show that the efficiency to attain a high low packing density of particles on the substrate is strongly dependent on the value of the exponent and on the characteristics of the dynamics

feasible optimality implies hack s law we analyze the elongation the scaling properties of drainage area with mainstream length in optimal channel networks ocns obtained through different algorithms searching for the minimum of a functional computing the total energy dissipation of the drainage system the algorithms have different capabilities to overcome the imprinting of initial and boundary conditions and thus they have different chances of attaining the global optimum we find that suboptimal shapes i e dynamically accessible states characterized by locally stationary total potential energy show the robust type of elongation that is consistently observed in nature this suggestive and directly measurable property is not found in the so called ground state i e the global minimum whose features including elongation are known exactly the global minimum is shown to be too regular and symmetric to be dynamically accessible in nature owing to features and constraints of erosional processes thus hack s law is seen as a signature of feasible optimality thus yielding further support to the suggestion that optimality of the system as a whole explains the dynamic origin of fractal forms in nature

johann faulhaber and sums of powers early 1 7th century mathematical publications of johann faulhaber contain some remarkable theorems such as the fact that the r fold summation of lm 2m nm is a polynomial in n n r when m is a positive odd number the present paper explores a computation based approach by which faulhaber may well have discovered such results and solves a 360 year old riddle that faulhaber presented to his readers it also shows that similar results hold when we express the sums in terms of central factorial powers instead of ordinary powers faulhaber s coefficients can moreover be generalized to noninteger exponents obtaining asymptotic series for 1 2 na in powers of n 1 n 1 1

definitions and properties for m concurrence designs summary incomplete block designs can be regarded as being in some sense approximately partially balanced in this paper we show that parameters pjk corresponding to those used for partially balanced designs can be defined for certain designs with m distinct concurrences these parameters have properties analogous to those for m associate class partially balanced incomplete block pbib m designs and have proved useful in searching for optimal designs this leads to a new upper bound for the efficiency factor which for given pjk values is attained if a pbib 2 design exists some results on complement and dual designs are also given

average efficiency factors and adjusted orthogonality in multidimensional designs summary an upper bound is given for the average efficiency factor of a connected multidimensional design it is shown that when the design has adjusted orthogonality its average efficiency factor is a simple function of the average efficiency factors of its component designs

perturbation theory for the laplacian on automorphic functions let f c psl 2 ir be a discrete subgroup with quotient f h of finite volume but not compact the spectrum of the laplacian on l2 auto morphic functions is unstable under perturbations however it becomes much more manageable when the scattering frequencies are adjoined with multiplic ity equal to the order of the pole of the determinant of the scattering matrix at these points this augmented set shows up in a natural way in a one sided version of the selberg trace formula and is the actual spectrum of the generator of a cut off wave equation applying standard perturbation theory to this op erator it is proved that the augmented spectrum is real analytic in teichmuller space the same operator is used to derive fermi s golden rule in this setting it turns out that the proper multiplicity to be attached to the laplacian eigen value at 4 is twice the dimension of cusp forms plus u tr f i 2 here 1 denotes the scattering matrix at this point it is shown that the generic value of u in the teichmuller space of the once punctured torus and the six times punctured sphere is zero this is also true of the x twisted spectral problem where x is a character for f department of mathematics stanford university stanford california 94305 department of mathematics princeton university princeton new jersey 08544 this content downloaded from 157 55 39 18 on sun 07 aug 2016 05 03 47 utc all use subject to http about jstor org terms

random matrix theory and discrete moments of the riemann zeta function we calculate the discrete moments of the characteristic polynomial of a random unitary matrix evaluated a small distance away from an eigenangle such results allow us to make conjectures about similar moments for the riemann zeta function and provide a uniform approach to understanding moments of the zeta function and its derivative

moments of the derivative of characteristic polynomials with an application to the riemann zeta function we investigate the moments of the derivative on the unit circle of characteristic polynomials of random unitary matrices and use this to formulate a conjecture for the moments of the derivative of the riemann function on the critical line we do the same for the analogue of hardy s z function the characteristic polynomial multiplied by a suitable factor to make it real on the unit circle our formulae are expressed in terms of a determinant of a matrix whose entries involve the i bessel function and alternately by a combinatorial sum

the fourth moment of ζ ρ discrete moments of the riemann zeta function were studied by gonek 3 4 and hejhal 8 in the 1980s they independently formulated a conjecture concerning the size of these moments in 1999 hughes keating and o connell 9 by employing a random matrix model made this conjecture more precise subject to the riemann hypothesis we establish upper and lower bounds of the correct order of magnitude in the case of the fourth moment

the construction of neighbour designs summary an algorithm is presented for the construction of efficient neighbour designs for use in field variety trials the method is based on a two stage procedure whereby a desirable a array is obtained first and then indexed to produce a neighbour array three schemes for allocating weights to plots within a block are compared

exponential sums and lattice points iii the gauss circle problem and the dirichlet divisor problem are special cases of the problem of counting the points of the integer lattice in a planar domain bounded by a piecewise smooth curve in the bombieri iwaniec mozzochi exponential sums method we must count the number of pairs of arcs of the boundary curve which can be brought into coincidence by an automorphism of the integer lattice these coincidences are parametrised by integer points close to certain plane curves the resonance curves this paper sets up an iteration step from a strong hypothesis about integer points close to curves to a bound for the discrepancy the number of integer points minus the area as in the latest work on single exponential sums the bombieri iwaniec mozzochi method itself gives bounds for the number of integer points close to a curve in part of the required range and it can in principle be used iteratively we use a bound obtained by swinnerton dyer s approximation determinant method in the discrepancy estimate o r k log r lambda in terms of the maximum radius of curvature r we reduce k from 2 3 classical and 46 73 paper ii in this series to 131 208 the corresponding exponent in the dirichlet divisor problem becomes k 2 131 416

deformations of maass forms we describe numerical calculations which examine the phillips sarnak conjecture concerning the disappearance of cusp forms on a noncompact finite volume riemann surface s under deformation of the surface our calculations indicate that if the teichmuller space of s is not trivial then each cusp form has a set of deformations under which either the cusp form remains a cusp form or else it dissolves into a resonance whose constant term is uniformly a factor of 10 8 smaller than a typical fourier coefficient of the form we give explicit examples of those deformations in several cases

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

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

mean values of derivatives of l functions in function fields i abstract we investigate the first moment of the second derivative of quadratic dirichlet l functions over the rational function field we establish an asymptotic formula when the cardinality of the finite field is fixed and the genus of the hyperelliptic curves associated to a family of dirichlet l functions over f q t tends to infinity as a more general result we compute the full degree three polynomial in the asymptotic expansion of the first moment of the second derivative of this particular family of l functions

nonresolvable incomplete block designs with few replicates summary tables are given showing incomplete block designs with up to 5 replicates for parameter values 10 v 60 2 r k 10 where r and k have a factor in common the designs are nonresolvable generalized cyclic incomplete block designs as defined by jarrett hall 1978 they give a general improvement in efficiency factor over the fractional cyclic incomplete block designs of john wolock david 1972 and provide a considerable extension to the tables of resolvable designs given by patterson williams hunter 1978

on the distribution of spacings between zeros of the zeta function etude numerique de la distribution des espacements des zeros de la fonction zeta de riemann observation de certains phenomenes inatendus

the riemann zeta function and its derivatives formulas for higher derivatives of the riemann zeta function are developed from ramanujan s theory of the constant of series by using the euler maclaurin summation methods formulas for zeta n s zeta n 1 s and zeta n 0 are obtained additional formulas involving the stieltjes constants are also derived analytical expression for error bounds is given in each case the formulas permit accurate derivative evaluation and the error bounds are shown to be realistic a table of zeta prime s is presented to 20 significant figures for s 20 0 1 20 for rational arguments zeta 1 k zeta prime 1 k are given for k 10 1 10 the first ten zeros of zeta prime s are also tabulated because the stieltjes constants appear in many formulas the constants were evaluated freshly for this work formulas for the gamma n are derived with new error bounds and a tabulation of the constants is given from n 0 to 100

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

higher power moments of δ x e t and p x let x denote the error term in the asymptotic formula for the divisor sum using a voronoi type formula for x we prove that 2 x x 3 dx cx 4 2 and 2 x x 4 dx c x 2 where c and c are some positive constants the estimate for the third power moment shows that 3 x is strongly biased towards the positive side we also obtain similar results for e t the error term in the mean square formula for the riemann zeta function on the critical line and for p x the error term in the circle problem

random matrix theory and the derivative of the riemann zeta function random matrix theory is used to model the asymptotics of the discrete moments of the derivative of the riemann zeta function s evaluated at the complex zeros i n we also discuss the probability distribution of ln 1 2 i n proving the central limit theorem for the corresponding random matrix distribution and analysing its large deviations

generalised incomplete trojan designs summary generalised incomplete m x n k trojan designs for m replicates of nk treatments based on sets of k cyclic generators are discussed normal equations for plots within columns plots within blocks and blocks within columns treatment effects are developed the nk treatments are divided into k subsets each of size n and the conditional plots within blocks and blocks within columns information matrix for each subset is defined efficient conditional treatment estimates are discussed and efficient generators for the various strata are discussed balanced m x n k incomplete trojan designs based on youden generators are constructed and designs based on multiples of a single generator are discussed some ideas for constructing efficient general m x n 2 designs are outlined and some advantages of generalised incomplete trojan designs are discussed

the average behaviour of the error term in a new kind of the divisor problem abstract we define a new kind of divisor function d 1 n by the nth coefficient of the dirichlet series s 2 and denote by 1 x the error term in the asymptotic formula for n x d 1 n then we compute the first moment of 1 x that is 1 x 1 x d x and consider discrete mean values n x 1 k n k 1 2 and deduce asymptotic formulas which are analogous to the results obtained by voronoi hardy and furuya