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

some bounds for the spectral radius of nonnegative tensors in this paper we extend the well known column sum bound of the spectral radius for nonnegative matrices to the tensor case an upper bound of the spectral radius for a nonnegative tensor is given via the largest eigenvalue of a symmetric tensor also we show some bounds of spectral radius of nonnegative tensors based on the sum of the entries in the other indices of tensors we demonstrate that our new results improve existing results the other main results of this paper is to provide a sharper ky fan type theorem and a comparison theorem for nonnegative tensors finally we make use of our bounds to give a perturbation bound for the spectral radius of symmetric nonnegative tensors this result is similar to the weyl theorem for the matrix case

z eigenpair bounds for an irreducible nonnegative tensor abstract in this paper we consider the z eigenpair of a tensor in particular an irreducible nonnegative tensor we present some bounds for the eigenvector and z spectral radius the proposed bounds improve some existing ones an example with practical applications is given to show the proposed bound

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

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

non vanishing of high derivatives of automorphic l functions at the center of the critical strip keywords modular l functions mollification nonvanishing results central value of high derivatives analytic rank jacobian variety reference tan article 2000 004doi 10 1515 crll 2000 074 record created on 2008 11 14 modified on 2017 05 12

weighted averages of modular l values using an explicit relative trace formula on gl 2 we derive a formula for averages of modular l values in the critical strip weighting by fourier coefficients hecke eigenvalues and petersson norms as an application we show that a grh holds for these averages as the weight or the level goes to we also use the formula to give explicit zero free regions of the form im s 0 for some particular modular l functions

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

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 mean values of mollifiers and l functions associated to primitive cusp forms we study the second moment of the l function associated to a holomorphic primitive cusp form of even weight perturbed by a new family of mollifiers this family is a natural extension of the mollifers considered by conrey and by bui conrey and young as an application we improve the current lower bound on critical zeros of holomorphic primitive cusp forms

time ordered operators i foundations for an alternative view of reality abstracr the purpose of this paper is to present the proper framework for the mathematical foundations of time ordered operators we introduce a new mathematical process which we call the chronological process this process generalizes the notion of a limit and allows us to recapture many properties lost in time ordering we then construct time ordered integrals and evolution operators we show that under reasonable assumptions the time ordered sum of two generators of contraction semigroups is a generator this result resolves a question that has been debated in physics for forty years

appendix an effective zero free region all the notation in this appendix will be as in the preceding paper let f be a maass form which is a newform for fo n with eigenvalue a and central character x normalized so that f f 1 we have seen that the size of p l the first fourier coefficient of f is intimately related to the behavior of l s f near s 1 here l s f is the l series of f the adjoint square lift of f to gl 3 and the crucial question is whether or not l s f vanishes when s is real and close to 1 it was shown in theorem 0 1 that if l s f is nonzero in a sufficiently wide neighborhood of 1 then ip 1 12 0 where the implied constant depends on e and is ineffective about a year after the preceding paper was first circulated it developed from conversations involving the above three authors that by slightly modifying the techniques introduced in that paper the possibility of a siegel zero could be completely eliminated in many cases in particular theorem 0 1 is now true unconditionally in the generic situation when f is not a lift from gl 1 that is to say when the l series of f is not equal to a hecke l series defined over a quadratic field if we include all cusp forms we can still obtain the theorem in the a aspect but must restrict ourselves to either an ineffective constant or a weaker effective constant in the n aspect we have

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

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

the tensor splitting with application to solve multi linear systems abstract in this paper firstly we introduce the variant tensor splittings and present some equivalent conditions for a strong m tensor based on the tensor splitting secondly the existence and uniqueness conditions of the solution for multi linear systems are given thirdly we propose some tensor splitting algorithms for solving multi linear systems with coefficient tensor being a strong m tensor as an application a tensor splitting algorithm for solving the multi linear model of higher order markov chains is proposed numerical examples are given to demonstrate the efficiency of the proposed algorithms

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

foundations for relativistic quantum theory i feynman s operator calculus and the dyson conjectures in this paper we provide a representation theory for the feynman operator calculus this allows us to solve the general initial value problem and construct the dyson series we show that the series is asymptotic thus proving dyson s second conjecture for quantum electrodynamics in addition we show that the expansion may be considered exact to any finite order by producing the remainder term this implies that every nonperturbative solution has a perturbative expansion using a physical analysis of information from experiment versus that implied by our models we reformulate our theory as a sum over paths this allows us to relate our theory to feynman s path integral and to prove dyson s first conjecture that the divergences are in part due to a violation of heisenberg s uncertainly relations

perron frobenius theorem for nonnegative multilinear forms and extensions we prove an analog of perron frobenius theorem for multilinear forms with nonnegative coefficients and more generally for polynomial maps with nonnegative coefficients we determine the geometric convergence rate of the power algorithm to the unique normalized eigenvector

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

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

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

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

sur le changement de signe des sommes de kloosterman combining sieve methods with automorphic form theory and techniques from l adic cohomology we prove that the sign of kloosterman sums kl 1 1 n changes infinitely often as n ranges over the squarefree integers having all their prime factors larger than n 1 23 9

pseudo spectra theory of tensors and tensor polynomial eigenvalue problems abstract this paper is devoted to the extension of the pseudo spectra theory from matrices to tensors based on the definition of an eigenpair of real symmetric tensors and results on the pseudo spectrum of square matrices we first introduce the pseudo spectrum of a complex tensor and investigate its fundamental properties such as its computational interpretations and the link with the stability of its related homogeneous dynamical system we then extend the pseudo spectrum to the setting of tensor polynomial eigenvalue problems we further derive basic structure of the pseudo spectrum for tensor polynomial eigenvalue problems including the symmetry boundedness and number of connected components under suitable mild assumptions finally we discuss the implication of the pseudo spectrum for computing the backward errors and the distance from a regular tensor polynomial to the nearest irregular tensor polynomial

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

brauer type eigenvalue inclusion sets and the spectral radius of tensors abstract in this paper we give two brauer type eigenvalue inclusion sets and some bounds on the spectral radius for tensors as applications some bounds on the spectral radius of uniform hypergraphs are presented

some new inequalities for the minimum h eigenvalue of nonsingular m tensors abstract this paper is devoted to deriving some new upper and lower bounds for the minimum h eigenvalue of irreducible nonsingular m tensors we demonstrate that the proposed bounds improve some existing ones and numerical examples are given to illustrate this fact additionally we extend these bounds to general nonsingular m tensors the other main results in this paper is to provide sharper ky fan theorem and comparison theorem for nonsingular m tensors which are better than the corresponding ones obtained by he and huang 2014 12

on the uniqueness and non uniqueness of the positive z eigenvector for transition probability tensors abstract we study the basic properties of transition probability tensors we investigate sufficient conditions for this class of tensors to ensure the uniqueness of the positive z eigenvector we use three different methods contraction mappings monotone operators and the brouwer index of fixed points

on l functions of elliptic curves and cyclotomic towers in this paper we shall strengthen a recent theorem of ralph greenberg 1 concerning l functions of elliptic curves and anticyclotomic towers let k be an imaginary quadratic field and m an abelian extension of k possibly of infinite degree we say that the extension m k is anticyclotomic if m is galois over q and if the nontrivial element of gal k q acts on gal m k by inversion now fix a finite set p of rational primes and consider the compositum in an algebraic closure of k of all anticyclotomic extensions of k which are unramified outside p we denote this compositum by l and refer to it as the maximal anticyclotomic extension of k unramified outside p the field l may also be described as the union of all ring class fields of k with conductor divisible only by primes in p class field theory shows that gal l k is isomorphic to the product of a finite group and the group

averages of central l values of hilbert modular forms with an application to subconvexity we use the relative trace formula to obtain exact formulas for central values of certain twisted quadratic base change l functions averaged over hilbert modular forms of a fixed weight and level we apply these formulas to the subconvexity problem for these l functions we also establish an equidistribution result for the hecke eigenvalues weighted by these l values

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

time ordered operators and feynman dyson algebras an approach to time ordered operators based upon von neumann s infinite tensor product hilbert spaces is used to define feynman dyson algebras this theory is used to show that a one to one correspondence exists between path integrals and semigroups which are integral operators defined by a kernel the reproducing property of the kernel being a consequence of the semigroup property for path integrals constructed from two semigroups the results are more general than those obtained by the use of the trotter kato formula perturbation series for the feynman dyson operator calculus for time evolution and scattering operators are discussed and it is pointed out that they are asymptotic in the sense of poincare as defined in the theory of semigroups thereby giving a precise formulation to a well known conjecture of dyson stated many years ago in the context of quantum electrodynamics moreover the series converge when these operators possess suitable holomorphy properties

the dirichlet series of koecher and maas and modular forms of weight 3 2 the basis problem for modular forms is the question which modular forms can be expressed as linear combinations of theta series in order that this question makes sense one has of course to specify which type of theta series should be taken into account we want here to follow in a special case the approach taken e g in b61 w a l l which in its general form asks which cuspidal modular forms f of degree n and weight k for the group foc n can be expressed as linear combinations of the theta series of degree n of integral quadratic forms of rank m 2 k and the same level n moreover one wants to derive an explicit expression for such a linear combination in terms of the petersson inner products of f with the theta series involved the representation theoretic version of this question is to ask which cuspidat automorphic representations of the adelic symplectic or metaplectic group

eigenvalues and invariants of tensors abstract a tensor is represented by a supermatrix under a co ordinate system in this paper we define e eigenvalues and e eigenvectors for tensors and supermatrices by the resultant theory we define the e characteristic polynomial of a tensor an e eigenvalue of a tensor is a root of the e characteristic polynomial in the regular case a complex number is an e eigenvalue if and only if it is a root of the e characteristic polynomial we convert the e characteristic polynomial of a tensor to a monic polynomial and show that the coefficients of that monic polynomial are invariants of that tensor i e they are invariant under co ordinate system changes we call them principal invariants of that tensor the maximum number of principal invariants of mth order n dimensional tensors is a function of m and n we denote it by d m n and show that d 1 n 1 d 2 n n d m 2 m for m 3 and d m n m n 1 m for m n 3 we also define the rank of a tensor all real eigenvectors associated with nonzero e eigenvalues are in a subspace with dimension equal to its rank

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

the subconvexity problem for rankin selberg l functions and equidistribution of heegner points ii we prove a general subconvex bound in the level aspect for rankin selberg l functions associated with two primitive holomorphic or maass cusp forms over q we use this bound to establish the equidistribution of incomplete galois orbits of heegner points on shimura curves associated with indefinite quaternion algebras over q

the square free sieve and the rank of elliptic curves let e be an elliptic curve over q a celebrated theorem of mordell asserts that e q the abelian group of rational points of e is finitely generated by the rank of e we mean the rank of e q thus the rank of e is positive if and only if e possesses an infinity of rational points relatively few general qualitative assertions can be made about the rank as e varies how large can the rank get although we expect that there are elliptic curves over q with arbitrarily high ranks this is presently unknown what is the average size of the rank recently brumer and mcguinness have reported on their study of 310 716 elliptic curves of prime conductor less than 108 where they have found that 20 06 of those curves have even rank 2 cf bm also see forthcoming work of brumer where subject to a number of standard conjectures he shows that 2 3 is an upper bound for the average rank for all elliptic curves over q ordered in terms of their faltings height what is the behavior of the rank over the family of twists of a given elliptic curve here one has three natural kinds of families of twists 1 quadratic twists one can take any elliptic curve and systematically twist it by all quadratic characters this is the type of family of elliptic curves we are concerned with in this paper specifically if e is an elliptic curve over q given by the weierstrass equation y2 x a x b and d is any squarefree integer the quadratic twist of e by d ed is given by the equation

the bounds on spectral radius of nonnegative tensors via general product of tensors abstract in 1 j y shao proposed the general product of tensors based on this product the minc type bound on spectral radius of nonnegative tensors is given and the generalized minimax theorem of nonnegative tensors is presented moreover we give some bounds on the spectral radius of the general product of tensors

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

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

time ordered operators ii apres avoir construit les fondements mathematiques generaux pour des equations d evolution lineaires en temps ordonne on applique les resultats pour montrer qu a la fois le developpement en perturbation et la methode des diagrammes de feymann sont mathematiquement sains

galois representations attached to q curves and the generalized fermat equation a 4 b 2 c p we prove that the equation a 4 b 2 c p has no solutions in coprime positive integers when p 211 the main step is to show that for all sufficiently large primes p every q curve over an imaginary quadratic field k with a prime of potentially multiplicative reduction greater than 6 has a surjective mod p galois representation the bound on p depends on k and the degree of the isogeny between e and its galois conjugate but is independent of the choice of e the proof of this theorem combines geometric arguments due to mazur momose darmon and merel with an analytic estimate of the average special values of certain l functions

symmetric nonnegative tensors and copositive tensors abstract we first prove two new spectral properties for symmetric nonnegative tensors we prove a maximum property for the largest h eigenvalue of a symmetric nonnegative tensor and establish some bounds for this eigenvalue via row sums of that tensor we show that if an eigenvalue of a symmetric nonnegative tensor has a positive h eigenvector then this eigenvalue is the largest h eigenvalue of that tensor we also give a necessary and sufficient condition for this we then introduce copositive tensors this concept extends the concept of copositive matrices symmetric nonnegative tensors and positive semi definite tensors are examples of copositive tensors the diagonal elements of a copositive tensor must be nonnegative we show that if each sum of a diagonal element and all the negative off diagonal elements in the same row of a real symmetric tensor is nonnegative then that tensor is a copositive tensor some further properties of copositive tensors are discussed

some bounds for h eigenpairs and z eigenpairs of a tensor abstract in this paper we consider the h eigenpairs and z eigenpairs of a tensor by estimating the ratio of the smallest and largest entries in the perron vector we present some sharper bounds for the h spectral radius of a nonnegative irreducible tensor similarly we also obtain some z spectral radius bounds of an irreducible weakly symmetric tensor in addition by using the technique of matricizing several bounds for the z spectrum are derived these proposed bounds improve some existing ones and some numerical examples are given to show the theoretical results

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