universal amplitude ratios for self avoiding walks on the kagome lattice we have calculated exactly the number the mean square end to end distance the mean square radius of gyration and the mean square distance of a monomer from the origin for n step self avoiding walks on the kagome lattice up to 30 steps we have estimated the connective constant and the critical amplitudes our numerical results are consistent with the theoretical prediction by cardy and saleur 1989 on the universality for certain amplitude ratios

self avoiding polygons on the square l and manhattan lattices transfer matrix techniques are used to extend the self avoiding polygon generating function on the square lattice to terms in x46 corresponding to 46 step polygons these techniques are then extended to apply to directed square lattices such as the l and manhattan lattice and the self avoiding polygon generating function to x48 is found for these lattices series analysis confirms that the specific heat exponent alpha 1 2 for the self avoiding walk problem and gives the following estimates for the connective constants mu sq 2 638155 or 0 000025 mu l 1 5657 or 0 0019 and mu man 1 7328 or 0 0005 some evidence for a correction to scaling exponent delta approximately 0 84 is found from square lattice series

universal distance ratios for 2d saws series results the mean square distance of a monomer to the origin and the mean square distance of the centre of mass to the origin are studied by exact enumeration on the square and triangular lattices the numerical results are inconsistent with the theoretical prediction of cardy and saleur 1989 by more than fifteen times the numerical uncertainty in the authors calculation

mean square values of hecke l series formed with r th order characters let l be a number field containing the r th roots of unity starting with the rankin selberg convolution of a metaplectic eisenstein series on the r fold cover of gl 2 with itself we construct a dirichlet series defined over l whose coefficients involve the r th order twists of a fixed hecke l function we then observe that a group of functional equations can be naturally associated with this construction combining this with the convexity theorem for holomorphic functions of several complex variables we show that this object as a function of two complex variables admits meromorphic continuation to 2 as an application we obtain asymptotic formulae for mean square values of the r th order twists of an arbitrary hecke l function defined over l

universal amplitude combinations for self avoiding walks polygons and trails the authors give exact relations for a number of amplitude combinations that occur in the study of self avoiding walks polygons and lattice trails in particular they elucidate the lattice dependent factors which occur in those combinations which are otherwise universal show how these are modified for oriented lattices and give new results for amplitude ratios involving even moments of the area of polygons they also survey numerical results for a wide range of amplitudes on a number of oriented and regular lattices and provide some new ones

constructing weyl group multiple dirichlet series let be a reduced root system of rank r a weyl group multiple dirichlet series for is a dirichlet series in r complex variables s1 sr initially converging for re si sufficiently large that has meromorphic continuation to c and satisfies functional equations under the transformations of c corresponding to the weyl group of a heuristic definition of such series was given in bbc06 and they have been investigated in certain special cases in bbc06 bbf06 bbfh07 bbf08 bbfa cg07 cfg08 cg chi05 in this paper we generalize results in cg to construct weyl group multiple dirichlet series by a uniform method and show in all cases that they have the expected properties

on kubota s dirichlet series kubota 19 showed how the theory of eisenstein series on the higher metaplectic covers of sl2 which he discovered can be used to study the analytic properties of dirichlet series formed with n th order gauss sums in this paper we will prove a functional equation for such dirichlet series in the precise form required by the companion paper 2 closely related results are in eckhardt and patterson 10 the kubota dirichlet series are the entry point to a fascinating universe their residues for example are mysterious if n 3 though there is tantalizing evidence that these residues exhibit a rich structure that can only be partially glimpsed at this time when n 4 the residues are the fourier coefficients of the biquadratic theta function that were studied by suzuki 23 suzuki found that he could only determine some of the coefficients this failure to determine all the coefficients was explained in terms of the failure of uniqueness of whittaker models for the generalized theta series by deligne 9 and by kazhdan and patterson 15 on the other hand patterson 22 conjectured that the mysterious coefficients are essentially square roots of gauss sums evidence for patterson s conjecture is discussed in bump and hoffstein 6 and in eckhardt and patterson 10 where the conjecture is refined in light of numerical data partial proofs were given by suzuki in 24 and 25 another set of conjectures relevant to the mysterious coefficients of n th order theta functions were given by bump and hoffstein who considered theta functions on the n fold covers of glr for arbitrary r they are expressed as identities between rankin selberg convolutions of generalized theta series and whittaker coefficients of eisenstein series on the metaplectic group but they boil down to properties of the residues of kubota dirichlet series and their higher rank generalizations see bump and hoffstein 6 bump 4 and hoffstein 12 these conjectures are different from the patterson conjecture and there are other considerations which suggest that there may be further unproved relations beyond those described in the conjectures of patterson and bump and hoffstein

universal amplitude combinations for self avoiding walks and polygons on directed lattices two directed lattices the l lattice and manhattan lattice are studied we have calculated exactly the number the mean square end to end distance the mean square radius of gyration and the mean square distance of a monomer from the origin for n step self avoiding walks on the l lattice and manhattan lattice for up to 60 and 50 steps respectively we have also computed the number and mean square radius of gyration for self avoiding polygons on the l lattice and manhattan lattice for up to 80 and 60 steps respectively we have estimated the critical amplitudes and our numerical results are consistent with the conjecture of universality for certain amplitude combinations however some amplitude combinations have values different from the corresponding values for undirected lattices

cubic twists of gl 2 automorphic l functions let k mathbb q sqrt 3 and let be a cuspidal automorphic representation of gl 2 mathbb a k consider the family of twisted l functions l s where ranges over the cubic hecke characters of k in this paper the mean value of this family of l functions is computed the result is consistent with the generalized lindelof hypothesis from this mean value result a nonvanishing theorem is established for given s there are infinitely many cubic twists such that the l value at s is nonzero at the center of the critical strip the number of such characters of norm less than x is gg x 1 2 epsilon these results are obtained by introducing and studying three different families of weighted double dirichlet series these series are related by functional equations some of which are obtained through the study of higher metaplectic eisenstein series and the hasse davenport relation the authors establish the continuation of such series and then obtain their main result by tauberian methods

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

weyl group multiple dirichlet series eisenstein series and crystal bases we show that the whittaker coecients of borel eisenstein series on the metaplectic covers of glr 1 can be described as multiple dirichlet series in r complex variables whose coecients are computed by attaching a number theoretic quantity a product of gauss sums to each vertex in a crystal graph these gauss sums depend on string data previously introduced in work of lusztig berenstein and zelevinsky and littelmann these data are the lengths of segments in a path from the given vertex to the vertex of lowest weight depending on a factorization of the long weyl group element into simple reections the coecients may also be described as sums over strict gelfand tsetlin patterns the description is uniform in the degree of the metaplectic cover

weyl group multiple dirichlet series for symmetrizable kac moody root systems weyl group multiple dirichlet series introduced by brubaker bump chinta friedberg and hoffstein are expected to be whittaker coefficients of eisenstein series on metaplectic groups chinta and gunnells constructed these multiple dirichlet series for all the finite root systems using the method of averaging a weyl group action on the field of rational functions in this paper we generalize chinta and gunnells work and construct weyl group multiple dirichlet series for the root systems associated with symmetrizable kac moody algebras and establish their functional equations and meromorphic continuation introduction weyl group multiple dirichlet series were introduced in the paper 3 by brubaker bump chinta friedberg and hoffstein and have been studied in their subsequent works 4 5 7 these multiple dirichlet series are defined for a finite root system of rank r and a number field f containing the 2n th roots of unity and unify many examples in number theory that have been studied previously on a case bycase basis and are expected to be whittaker coefficients of eisenstein series on metaplectic groups this expectation is now called an eisenstein conjecture and the conjecture has been proven for the root systems of type ar 6 moreover the theory of weyl group multiple dirichlet series can be applied to the moments problem of the riemann zeta function and dirichlet l functions a nice survey on this subject can be found in 9 it is remarkable that there are two distinct constructions of these multiple dirichlet series in the work of brubaker bump and friedberg 5 6 the local coefficients are defined using the data from kashiwara s crystal graph 14 more precisely each local coefficient is given as a sum of g v over the crystals v of the same weight where g v is a product of gauss sums that are determined by the crystal graph the construction has been completed for most of the classical root systems the second construction is due to chinta and gunnells 10 11 they defined a new weyl group action on the field of rational functions in several variables the action involves gauss sums and works for all the finite root systems in a coherent way by taking an average of the weyl group action they obtained a deformed weyl character and defined the local coefficients using the deformed character the two constructions are expected to be equivalent and the equivalence in type a has been established 6 12 16 received by the editors october 10 2012 and in revised form october 12 2012 november 2 2012 april 11 2013 and april 21 2013 2010 mathematics subject classification primary 11f68 secondary 17b67 c 2014 american mathematical society reverts to public domain 28 years from publication

cycle integrals of the j function and mock modular forms in this paper we construct certain mock modular forms of weight 1 2 whose fourier coecients are given in terms of cycle integrals of the modular j function their shadows are weakly holomorphic forms of weight 3 2 these new mock modular forms occur as holomorphic parts of weakly harmonic maass forms we also construct a generalized mock modular form of weight 1 2 having a real quadratic class number times a regulator as a fourier coecient as an application of these forms we study holomorphic modular integrals of weight 2 whose rational period functions have poles at certain real quadratic integers the fourier coecients of these modular integrals are given in terms of cycle integrals of modular functions such a modular integral can be interpreted in terms of a shimura type lift of a mock modular form of weight 1 2 and yields a real quadratic analogue of a borcherds product

bounds on self avoiding walks on directed square lattices the manhattan lattice is the covering lattice of the l lattice and the author has studied self avoiding walks on this pair of lattices upper and lower bounds on the connective constant mu have been obtained as have series analysis estimates

polynomial hulls with convex fibers and complex geodesics let x be a compact subset of z 1 cn with convex fibers several equivalent conditions are obtained that characterize polynomially convex hulls y x with nonempty interior under the latter assumption it is shown that every z0 w0 y and such that z0 1 is contained in a closed n dimensional complex submanifold of d cn which is tangent to y along an analytic disc we show as an application that some results of lempert on complex geodesies in convex domains are direct consequences of properties of polynomial hulls

on bounded analytic functions in finitely connected domains a new proof of the corona theorem for finitely connected domains is given it is based on a result on the existence of a meromorphic selection from an analytic set valued function the latter fact is also applied to the study of finitely generated ideals of h over multiply connected domains introduction in this paper which is a direct continuation of 18 we study applications of analytic multifunctions to some topics in function theory on finitely connected domains related mainly to the corona problem concerniag analytic multifunctions which are certain set valued functions cf definition 1 2 1 3 of 18 form sufficient background for our purposes here the reader is also referred there for the information about the origin of the approach employed in this paper however basic definitions and references are provided below our basic new result theorem 1 4 associates to every analytic multifunction defined in a finitely connected domain some meromorphic functions with poles at the critical points of green s function of this domain some improvements of this fact in txhe special case of an annulus are discussed in 3 this main technical result is applied in 2 to help obtain a new simple proof of the well known corona theorem for finitely connected domains a novel feature of this proof as compared with e g forelli 9 stout 19 gamelin 7 is that it does not use the corresponding result for the unit disc the same methods are used in 4 where we consider the question when some power of g from h g belongs to the ideal af1 af2 afn we obtain generalizations to multiply connected domains of a result due to t wolff theorems 4 1 and 4 2 as well as its refinements in the case of the unit disc corollary 4 4 and example 5 3 all proofs given in this paper extend to finite riemann surfaces 1 a meromorphic selection theorem the main result of this section relates some meromorphic vector valued functions to analytic multifunctions defined in finitely connected domains first we recall some definitions definition 1 1 folk a locally compact set z c cn is a maximum set if for every compact set n c z and for every analytic function f defined in a neighborhood of n the inequality maxn if l maxazn if l holds definition 1 2 16 18 anuppersemicontinuousset valuedcorrespondence z k z g 2c where g c c is open and all k z are nonempty and received by the editors january 16 1986 presented to the society special session on convex analysis new orleans january 11 1986 1980 mathematics subject classifiratson 1985 ren primary 32a80 32e20 46j15 2 1987 american mathematical society 0002 9947 87 1 00 25 per page

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

prehomogeneous vector spaces and field extensions letk be an infinite field of characteristic not equal to 2 3 5 in this paper we construct a natural map from the set of orbits of certain prehomogeneous vector spaces to the set of isomorphism classes of galois extensions ofk which are splitting fields of equations of certain degrees and prove that the inverse image of this map corresponds bijectively with conjugacy classes of galois homomorphisms

a spectral proof of class number one we continue our previous work on the subject of re proving the heegner baker stark theorem giving another effective resolution of this conjecture of gauss namely there are exactly 9 imaginary quadratic fields mathbf q sqrt q with class number 1 specifically the list is q in 3 4 7 8 11 19 43 67 163 we again follow an ansatz of goldfeld considering a modular form l function of analytic rank 2 and reducing the situation to showing sufficient cancellation of the dirichlet series coefficients when restricted to the principal form previously we then chose a specific rank 2 elliptic curve with complex multiplication by mathbf q sqrt 1 and deduced the desired cancellation from an equi distribution result of hooley s for roots of a quadratic polynomial to varying moduli herein we instead use spectral techniques filling in details of work of templier and tsimerman to complete the proof relying on the duke iwaniec bound for fourier coefficients of half integral weight maass forms unlike the previous method we do not require complex multiplication and any analytic rank 2 modular form whose level has an even number of prime factors suffices

weyl group multiple dirichlet series ii the stable case to each reduced root system of rank r and each sufficiently large integer n we define a family of multiple dirichlet series in r complex variables whose group of functional equations is isomorphic to the weyl group of the coefficients in these dirichlet series exhibit a multiplicativity that reduces the specification of the coefficients to those that are powers of a single prime p for each p the number of nonzero such coefficients is equal to the order of the weyl group and each nonzero coefficient is a product of n th order gauss sums the root system plays a basic role in the combinatorics underlying the proof of the functional equations

a parallel algorithm for the enumeration of self avoiding polygons on the square lattice we have developed a parallel algorithm that allows us to enumerate the number of self avoiding polygons on the square lattice to perimeter length 110 we have also extended the series for the first 10 area weighted moments and the radius of gyration to 100 analysis of the resulting series yields very accurate estimates of the connective constant 2 638 158 530 31 3 biased and the critical exponent 0 500 000 1 2 unbiased in addition we obtain very accurate estimates for the leading amplitudes confirming to a high degree of accuracy various predictions for universal amplitude combinations

universal distance ratios for two dimensional polymers by using a quantitative version of the c theorem in conformal theories the authors determine some universal geometrical features of two dimensional critical systems with emphasis on the ratios of mean square distances for polymers

interpolation and extrapolation of analytic multivalued functions on demontre des theoremes pour l interpolation et l extrapolation des fonctions multivaluees analytiques

universal distance ratios for 2d saws monte carlo and exact series results the mean square distance of a monomer to the origin and the mean square distance of the centre of mass to the origin are studied by monte carlo and exact series methods on the square and lattice the numerical results give strong evidence of a discrepancy with the theoretical prediction of cardy and saleur 1989 based on conformal invariance theory

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

on the density of unnormalized tamagawa numbers of orthogonal groups ii this is the second part of a series of three papers in this series of papers we determine the density of unnormalized tamagawa numbers of projective special orthogonal groups defined over a fixed number field in this part we determine local constants at infinite places which appear in the density theorem and prove the density theorem when the number of variables n is odd for that purpose we make a dirichlet series whose coefficients are unnormalized tamagawa numbers of projective special orthogonal groups and prove that it is rather simple when n is odd

a local riemann hypothesis i daniel bump1 kwok kwong choi2 par kurlberg3 jeffrey vaaler4 1 department of mathematics stanford university stanford ca 94305 2125 usa e mail bump math stanford edu 2 department of mathematics the university of hong kong pokfulam road hong kong hong kong e mail choi maths hku hk 3 department of mathematics university of georgia athens ga 30602 usa e mail kurlberg math uga edu 4 university of texas at austin department of mathematics rlm 8 100 austin tx 78712 1082 usa e mail vaaler math utexas edu

universal amplitude combinations for self avoiding polygons on the kagome lattice abstract we have calculated exactly the number the mean square radius of gyration and moments of area for n step self avoiding polygons on the kagome lattice up to 36 steps we estimated the critical amplitudes and our numerical results are consistent with the theoretical predictions by the universality for certain amplitude combinations

complex interpolation of normed and quasinormed spaces in several dimensions ii properties of harmonic interpolation this paper is a continuation of the study of harmonic interpolation families of normed or quasinormed spaces parametrized by points of a domain in ck it is shown among other things that each of the following properties holds for all the intermediate quasinormed spaces if it holds for all given boundary spaces 1 being a normed space 2 being a hilbert space 3 satisfying the triangle inequality by the rth power of the quasinorm 4 being uniformly convex and 5 being uniformly smooth as a principal tool the notion of a harmonic set valued function a generalization of analytic multifunction is introduced and studied

weyl group multiple dirichlet series iii eisenstein series and twisted unstable ar weyl group multiple dirichlet series were associated with a root system and a number field f containing the n th roots of unity by brubaker bump chinta friedberg and hostein 3 and brubaker bump and friedberg 4 provided n is suciently large their coecients involve n th order gauss sums the case where n is small is harder and is addressed in this paper when ar twisted dirichet series are considered which contain the series of 4 as a special case these series are not euler products but due to the twisted multiplicativity of their coecients they are determined by their p parts the p part is given as a sum of products of gauss sums parametrized by strict gelfand tsetlin patterns it is conjectured that these multiple dirichlet series are whittaker coecients of eisenstein series on the n fold metaplectic cover of glr 1 and this is proved if r 2 or n 1 the equivalence of our definition with that of chinta 11 when n 2 and r 6 5 is also established

a novel method of calculating amplitude ratios by series expansions the author introduces a new method of calculating critical amplitude ratios using series which is both simple and powerful this method which gives estimates for the amplitude ratios that are neither biased by the values of the critical points nor by the critical exponents is applied to several models it is shown that this method produces results where no reliable estimates from series expansion exist in particular one finds 0 025 or 0 001 for at b2 for the 3d ising model and 220 or 10 for c c for the two dimensional percolation model in agreement with and with more accuracy than values obtained by other methods

artin schreier extensions of the rational function field in this paper we study the arithmetic of artin schreier extensions of mathbb f q t we determine the integral closure of mathbb f q t in artin schreier extension of mathbb f q t we also investigate the average values of the l functions of orders of artin schreier extensions and study the average values of ideal class numbers when p 3 in detail

tests of hyperuniversality for self avoiding walks a universal combination of amplitudes for the end to end distance of n step self avoiding walks and for the number of n step self avoiding polygons is estimated numerically for various lattices in d 2 universality of this amplitude combination is confirmed to good accuracy while in d 3 the data are consistent with universality but the error limits are rather large for finite width square lattice strips the authors calculate universal critical point correlation amplitudes by fixed fugacity monte carlo simulations in addition these amplitudes are calculated analytically either exactly or approximately by cardy s conformal mapping approach numerical values agree well with the theoretical predictions

a metaplectic casselman shalika formula for gl r we provide formulas for various bases of spherical whittaker functions on the n fold metaplectic cover of rm gl r over a p adic field and show that there is a basis of symmetric functions in the complex parameter in addition we relate a specific spherical whittaker function to the p power part of the weyl group multiple dirichlet series for the root system of type a r 1 constructed from n th order gauss sums we also show that the zonal spherical functions can be computed explicitly in terms of hall littlewood polynomials as in macdonald s formula for rm gl r

double dirichlet series and the n th order twists of hecke l series let n 3 be a fixed integer and let f be a global field containing the n th roots of unity in this paper we study the collective behavior of the n th order twists of a fixed hecke l series for f to do so we introduce a double dirichlet series in two complex variables s w which is a weighted sum of the twists and obtain its meromorphic continuation we also study related sums of n th order gauss sums these objects together satisfy a nonabelian group of functional equations in s w of order 32

a converse theorem for double dirichlet series we prove that a certain vector valued double dirichlet series satisfying appropriate functional equations is a mellin transform of a vector valued metaplectic eisenstein series we establish an analogous result for scalar double dirichlet series of the type studied by siegel

asymptotic behaviors of class number sums associated with pell type equations it is well known that the pell equation t 2 du 2 4 has infinitely many integer solutions t u for a given d 0 with d equiv 0 1 bmod 4 and d ne l 2 for any integer l however for a square free integer n ne 1 the equation t 2 du 2 4n does not always have integer solutions and verifying its solubility insolubility is not easy in the present paper we propose the asymptotic formulas of the sum of the class numbers h d of the primitive indefinite binary quadratic forms over the discriminants d 0 for which the pell type equation t 2 du 2 4n has an integer solution to study the distributions of such d

first moment of hecke l functions with quartic characters at the central point in this paper we study the first moment of central values of hecke l functions associated with quartic characters

on the p parts of quadratic weyl group multiple dirichlet series let be a reduced root system of rank r a weyl group multiple dirichlet series for is a dirichlet series in r complex variables s1 sr initially converging for r si sufficiently large which has meromorphic continuation to c and satisfies functional equations under the transformations of c corresponding to the weyl group of two constructions of such series are available one 1 4 based on summing products of n th order gauss sums the second 5 based on averaging a certain group action over the weyl group in each case the essential work occurs at a generic prime p the local factors satisfying local functional equations are then pieced into a global object in this paper we study these constructions and the relationship between them first we extend the averaging construction to obtain twisted weyl group multiple dirichlet series whose p parts are given by evaluating certain rational functions in r variables then we develop properties of such a rational function giving its precise denominator showing that the nonzero coefficients of its numerator are indexed by points that are contained in a certain convex polytope determining the coefficients corresponding to the vertices and showing that in the untwisted case the rational function is uniquely determined from its polar behavior and the local functional equations we also give evidence that in the case ar the p part obtained here exactly matches the p part of the twisted multiple dirichlet series introduced in 3 when n 2

mean values of biquadratic zeta functions let h d be the number of primitive inequivalent binary quadratic forms of discriminant d in disquisitiones 302 and 304 8 gauss gives conjectures for the average value of h d if d 0 where in the latter case d is derived from the fundamental solution to pell s equation gauss s conjectures were first proved by lipschitz 12 for negative discriminants and siegel for positive discriminants 14 when d is a fundamental discriminant and d the primitive real dirichlet character associated to the quadratic extension q d dirichlet s class number formula relates the value of the l series of d at s 1 to the class number h d thus gauss s conjectured asymptotics are equivalent to conjectures for sums of the type