high precision computation of riemann s zeta function by the riemann siegel formula i we present rigorous and sharp bounds for the terms and remainder in the riemann siegel formula for a general argument not necessarily on the critical line this allows for the computation of s and z t to high precision we also derive the riemann siegel formula in a new and more direct way

duhamel s principle for temporally inhomogeneous evolution equations in banach space on precise les conditions sous lesquelles le principe de duhamel est valable pour des equations d evolution de type hyperbolique dans un espace de banach

geodesics and commensurability classes of arithmetic hyperbolic 3 manifolds this sharpens 10 where it was shown that the complex length spectrum of m determines its commensurability class suppose m is an arithmetic hyperbolic 3 manifold which is not commensurable to m theorem 1 1 implies ql m 6 ql m though by example 2 1 below it is possible that one of ql m or ql m contains the other by the length formulas recalled in 2 1 and 2 2 each element of ql m ql m is a rational multiple of the logarithm of a real algebraic number as noted by prasad and rapinchuk in 9 the gelfond schneider theorem 1 implies that a ratio of such logarithms is transcendental if it is irrational thus if ql m ql m then is transcendental for all non zero ql m recently prasad and rapinchuk have shown in 9 that if m is an arithmetic hyperbolic manifold of even dimension then ql m and the commensurability class of m determine one another in addition they have shown that this is not always true for arithmetic hyperbolic 5 manifolds however they have announced a proof that for all locally symmetric spaces associated to a specified absolutely simple lie group there are only finitely many commensurability classes of arithmetic lattices giving rise to a given rational length spectrum it is known see 4 pp 415 417 that for closed hyperbolic manifolds the spectrum of the laplace beltrami operator action on l2 m counting multiplicities determines the set of lengths of closed geodesics on m without counting multiplicities hence theorem 1 1 implies

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

necessary condition on the dissipative term for the asymptotically free property of dissipative kirchhoff equations abstract it is known that the solutions of linear abstract dissipative wave equation with integrable coefficient of the dissipation term are asymptotically free namely the solutions have the same asymptotic behavior as that of the solutions of an abstract free wave equation in this paper we conclude that the integrability is not enough for asymptotic free property of kirchhoff equation by showing the following for the existence of a non trivial asymptotically free solution of abstract kirchhoff equation with initial data in some class an essentially stronger decay condition on the coefficient of dissipation term is necessary

newton cotes integration for approximating stieltjes generalized euler constants in the laurent expansion s a 1 s 1 k 0 1 k k a k s 1 k 0 1 we also conjecture that limk k 1 2 k k 0 a special case of a more general conjecture relating the values of k a and k a for 0 0 we conjecture that for nonintegral r r is precisely 1 r times the r th weyl fractional derivative at s 1 of the entire function s 1 s 1 1 we also conjecture that g now defined for all real arguments r 0 is smooth our numerical method uses newton cotes integration formulae for very high degree interpolating polynomials it differs in implementation from but compares in error bounding to euler maclaurin summation based methods

homogenization and corrector for the wave equation with discontinuous coefficients in time abstract in this paper we analyze the homogenization of the wave equation with bounded variation coefficients in time generalizing the classical result which assumes lipschitz continuity we start showing a general existence and uniqueness result for a general sort of hyperbolic equations then we obtain our homogenization result comparing the solution of a sequence of wave equations to the solution of a sequence of elliptic ones we conclude the paper making an analysis of the corrector firstly we obtain a corrector result assuming that the derivative of the coefficients in the time variable is equicontinuous this result was known for non time dependent coefficients after we show with a counterexample that the regularity hypothesis for the corrector theorem is optimal in the sense that it does not hold if the time derivative of the coefficients is just bounded

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

new results on the stieltjes constants asymptotic and exact evaluation abstract the stieltjes constants k a are the expansion coefficients in the laurent series for the hurwitz zeta function about s 1 we present new asymptotic summatory and other exact expressions for these and related constants

nonlocal problems for boussinesq equations abstract in this paper the existence and uniqueness of solution of the integral boundary value problem for abstract boussinesq equations are obtained

explicit upper bounds for the remainder term in the divisor problem we first report on computations made using the gp pari package that show that the error term x in the divisor problem is mathscr m x 4 o 0 35 x 1 4 log x when x ranges 1 081 080 10 10 where mathscr m x 4 is a smooth approximation the remaining part and in fact most of the paper is devoted to showing that delta x le 0 397 x 1 2 when x ge 5 560 and that delta x le 0 764 x 1 3 log x when x ge 9 995 several other bounds are also proposed we use this results to get an improved upper bound for the class number of a quadractic imaginary field and to get a better remainder term for averages of multiplicative functions that are close to the divisor function we finally formulate a positivity conjecture concerning

small isospectral and nonisometric orbifolds of dimension 2 and 3 revisiting a construction due to vigneras we exhibit small pairs of orbifolds and manifolds of dimension 2 and 3 arising from arithmetic fuchsian and kleinian groups that are laplace isospectral in fact representation equivalent but nonisometric

scattering theory for temporally inhomogeneous hamilton equations in hilbert space in the scattering theory for the equation y p t y 0 in banach space x it is known that for a perturbation which is summable in time the scattering operator is everywhere defined in the space of the generalized solutions and onto we prove here that a stronger notion of scattering operator defined in the space of the true solutions is everywhere defined and onto provided that x is reflexive and that p satisfies the following property if p denotes the average operator p t tt 1p s ds then the cauchy problem for the equation y p t y 0 is uniformly well posed on the real line this result is applied to the hamilton equation in hilbert space jx h t x generalizing those of the author arch rational mech anal 86 1984 147 180 for the abstract wave equation u a t u 0

linear second order differential equations in hilbert spaces the cauchy problem and asymptotic behaviour for large time on considere des equations d evolutions lineaires d ordre 2 dans un espace de hilbert ceci fournit un schema abstrait pour l etude des equations aux derivees partielles hyperboliques a coefficients qui peuvent etre discontinus par rapport aux variables de temps et d espace on trouve l existence d une solution unique continue pour toutes les valeurs positives du temps

counting and effective rigidity in algebra and geometry the purpose of this article is to produce effective versions of some rigidity results in algebra and geometry on the geometric side we focus on the spectrum of primitive geodesic lengths resp complex lengths for arithmetic hyperbolic 2 manifolds resp 3 manifolds by work of reid this spectrum determines the commensurability class of the 2 manifold resp 3 manifold we establish effective versions of these rigidity results by ensuring that for two incommensurable arithmetic manifolds of bounded volume the length sets resp the complex length sets must disagree for a length that can be explicitly bounded as a function of volume we also prove an effective version of a similar rigidity result established by the second author with reid on a surface analog of the length spectrum for hyperbolic 3 manifolds these effective results have corresponding algebraic analogs involving maximal subfields and quaternion subalgebras of quaternion algebras to prove these effective rigidity results we establish results on the asymptotic behavior of certain algebraic and geometric counting functions which are of independent interest

explicit zero density estimate for the riemann zeta function near the critical line abstract in 1946 a selberg proved n t t 1 1 4 1 2 log t where n t is the number of nontrivial zeros of the riemann zeta function with and 0 t we provide an explicit version of this estimate together with an explicit approximate functional equation and an explicit upper bound for the second power moment of the zeta function on the critical line

abstract schroedinger type differential equations with variable domain abstract the cauchy problem is studied for a class of linear abstract differential equations of schroedinger type with variable domain existence and uniqueness results are proved for suitably defined weak solutions some applications are given they concern initial boundary value problems for linear schroedinger type p d e either with mixed variable lateral conditions or in non cylindrical regions

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

exponential sums and the riemann zeta function v a van der corput exponential sum is s sigma exp 2 pi i f m where m has size m the function f x has size t and alpha log m log t 1 there are different bounds for s in different ranges for alpha in the middle range where alpha is near 1 over 2 s o sqrt m t theta epsilon this theta bounds the exponent of growth of the riemann zeta function on its critical line rm re s 1 over 2 van der corput used an iteration which changed alpha at each step the bombieri iwaniec method whilst still based on mean squares introduces number theoretic ideas and problems the second spacing problem is to count the number of resonances between short intervals of the sum when two arcs of the graph of y f x coincide approximately after an automorphism of the integer lattice in the previous paper in this series proc london math soc 3 66 1993 1 40 and the monograph area lattice points and exponential sums we saw that coincidence implies that there is an integer point close to some resonance curve one of a family of curves in some dual space now calculated accurately in the paper resonance curves in the bombieri iwaniec method which is to appear in funct approx comment math we turn the whole bombieri iwaniec method into an axiomatised step an upper bound for the number of integer points close to a plane curve gives a bound in the second spacing problem and a small improvement in the bound for s ends and cusps of resonance curves are treated separately bounds for sums of type s lead to bounds for integer points close to curves and another branching iteration luckily swinnerton dyer s method is stronger we improve theta from 0 156140 in the previous paper and monograph to 0 156098 in fact 32 205 epsilon 269 410 epsilon is an exponent pair for every epsilon 0

global exponential stability and global attractivity of impulsive hopfield neural networks with time delays in this paper by means of constructing the extended impulsive delayed halanay inequality and by lyapunov functional methods we analyze the global exponential stability and global attractivity of impulsive hopfield neural networks with time delays some new sufficient conditions ensuring exponential stability of the unique equilibrium point of impulsive hopfield neural networks with time delays are obtained those conditions are more feasible than that given in the earlier references to some extent some numerical examples are also discussed in this work to illustrate the advantage of the results we obtained

spectrally similar incommensurable 3 manifolds reid has asked whether hyperbolic manifolds with the same geodesic length spectrum must be commensurable building toward a negative answer to this question we construct examples of hyperbolic 3 manifolds that share an arbitrarily large portion of the length spectrum but are not commensurable more precisely for every n 0 we construct a pair of incommensurable hyperbolic 3 manifolds nn and nn whose volume is approximately n and whose length spectra agree up to length n both nn and nn are built by gluing two standard submanifolds along a complicated pseudo anosov map ensuring that these manifolds have a very thick collar about an essential surface the two gluing maps differ by a hyper elliptic involution along this surface our proof also involves a new commensurability criterion based on pairs of pants

an improved upper bound for the argument of the riemann zeta function on the critical line this paper concerns the function s t the argument of the rie mann zeta function along the critical line the main result is that s t 0 111logt 0 275 log logt 2 450 which holds for all te

explicit zero density for the riemann zeta function abstract let n t denote the number of nontrivial zeros of the riemann zeta function with real part greater than and imaginary part between 0 and t we provide explicit upper bounds for n t commonly referred to as a zero density result in 1937 ingham showed the following asymptotic result n t o t 8 3 1 log t 5 ramare recently proved an explicit version of this estimate we discuss a generalization of the method used in these two results which yields an explicit bound of a similar shape while also improving the constants

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

numerical computations concerning the erh this paper describes a computation which established the erh to height 10000 for all primitive dirichlet l series with conductor q 13 and to height 2500 for all q 72 all composite q 112 and other moduli the computations were based on euler maclaurin summation care was taken to obtain mathematically rigorous results the zeros were first located within 10 12 then rigorously separated using an interval arithmetic package a generalized turing criterion was used to show there were no zeros off the critical line statistics about the spacings between zeros were compiled to test the pair correlation conjecture and gue hypothesis

explicit estimates on several summatory functions involving the moebius function we prove that d x d d log x 1 69 when x 96 955 and deduce from that d x d q 1 d d log x q 4 5 q q for every x q 1 we also give better constants when x q is larger furthermore we prove that 1 d x d log x d d 3 14 log x and several similar bounds from which we also prove corresponding bounds when summing the same quantity but with the additional condition d q 1 we prove similar results for d x d log x d d among which we mention the bound d x d log x d d 2 log x 2 0 5 24 log x where 0 is the euler constant we complete this collection by bounds such as d x d q 1 d x q q log x q we also provide all these bounds with variations where 1 log x is replaced by 1 1 log x

the riemann siegel expansion for the zeta function high orders and remainders on the critical line s i t t real riemann s zeta function can be calculated with high accuracy by the riemann siegel expansion this is derived here by elementary formal manipulations of the dirichlet series it is shown that the expansion is divergent with the high orders r having the familiar 9factorial9 divided by power9 dependence decorated with an unfamiliar slowly varying multiplier function which is calculated explicitly terms of the series decrease until r r 2 t and then increase the form of the remainder when the expansion is truncated near r is determined it is of order exp t indicating that the critical line is a stokes line for the riemann siegel expansion these conclusions are supported by computations of the first 50 coefficients in the expansion and of the remainders as a function of truncation for several values of t