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

where is the nearest non regular pencil abstract this is a first step toward the goal of finding a way to calculate a smallest norm deregularizing perturbation of a given square matrix pencil minimal de regularizing perturbations have geometric characterizations that include a variable projection linear least squares problem and a minimax characterization reminiscent of the courant fischer theorem the characterizations lead to new computationally attractive upper and lower bounds we give a brief survey and illustrate strengths and weaknesses of several upper and lower bounds some of which are well known and some of which are new the ultimate goal remains elusive

on the factorization of consecutive integers a classical result of sylvester 21 see also 16 17 generalizing bertrand s postulate states that the greatest prime divisor of a product of k consecutive integers greater than k exceeds k more recent work in this vein well surveyed in 18 has focussed on sharpening sylvester s theorem or upon providing lower bounds for the number of prime divisors of such a product as noted in 18 a basic technique in these arguments is to make a careful distinction between small and large primes and then apply sophisticated results from multiplicative number theory along these lines if we write n k u v n 2k

on the stability radius of a generalized state space system the concept of distance to instability of a system matrix is generalized to system pencils which arise in descriptor semistate systems difficulties arise in the case of singular systems because the pencil can be made unstable by an infinitesimal perturbation it is necessary to measure the distance subject to restricted or structured perturbations in this paper a suitable measure for the stability radius of a generalized state space system is defined and a computable expression for the distance to instability is derived for regular pencils of index less than or equal to one for systems which are strongly controllable it is shown that this measure is related to the sensitivity of the poles of the system over all feedback matrices assigning the poles

the average amount of information lost in multiplication we show that if x and y are integers independently and uniformly distributed in the set 1 n then the information lost in forming their product which is given by the equivocation h x y xmiddoty is theta loglogn we also prove two extremal results regarding cases in which x and y are not necessarily independently or uniformly distributed first we note that the information lost in multiplication can of course be 0 we show that the condition h x y xmiddoty 0 implies 2log2n h x y omega loglogn furthermore if x and y are independent and uniformly distributed on disjoint sets of primes it is possible to have h x y xmiddoty 0 with log2n h x and log2n h y each o loglogn second we show that no matter how x and y are distributed h x y xmiddoty o logn loglogn furthermore there are distributions in which x and y are independent and uniformly distributed over sets of numbers having only small and distinct prime factors for which we have h x y xmiddoty omega logn loglogn

estimates of theta x k l for large values of x we extend a result of ramare and rumely 1996 about the chebyshev function in arithmetic progressions we find a map e x such that x k l x k 0 whereas e x is a constant now we are able to show that for x 1531 x 3 l x 2 x 2 ln x

short effective intervals containing primes in arithmetic progressions and the seven cubes problem for any e 0 and any non exceptional modulus q 3 we prove that for x large enough x e log 2 q the interval e x e 3x e contains a prime p in any of the arithmetic progressions modulo q we apply this result to establish that every integer n larger than exp 71000 is a sum of seven cubes

probability laws related to the jacobi theta and riemann zeta functions and brownian excursions this paper reviews known results which connect riemann s integral representations of his zeta function involving jacobi s theta function and its derivatives to some particular probability laws governing sums of independent exponential variables these laws are related to one dimensional brownian motion and to higher dimensional bessel processes we present some characterizations of these probability laws and some approximations of riemann s zeta function which are related to these laws

an improved algorithm for the computation of kronecker s canonical form of a singular pencil we give an 0 t algorithm for computing the kronecker structure of an arbitrary m x n pencil xe a the algorithm is shown to he numerically stable because only unitary transformations are used the improved speed over earlier unitary methods is due to the efficient use of condensed forms which are maintained throughout the recursions of the algorithm

residuation of linear series and the effective cone of cd we obtain new information about divisors on the dth symmetric power cd of a general curve c of genus g 4 this includes a complete description of the effective cone of cg 1 and a partial computation of the volume function on one of its non nef subcones as well as new bounds for the effective and movable cones of cd in the range g 1 d g 2 we also obtain for each g 5 a divisor on cg 1 with non equidimensional stable base locus for a general hyperelliptic curve c of genus g we obtain a complete description of the effective cone of cd for 2 d g and an integral divisor on cg 1 which has non integral volume whenever g is not a power of 2 1 introduction let c be a smooth complex projective algebraic curve and let d 1 be an integer the dth symmetric power cd c d sd is a smooth d dimensional complex projective variety which is a fine moduli space parametriz ing effective divisors of degree d on c in this paper we study the cone of effective divisors on cd and its refinements there are two natural divisor classes on cd for each p c the image of the embedding

robust pole assignment in singular control systems abstract necessary conditions are given for the problem of pole assignment by state feedback in singular linear systems descriptor systems to have a solution which is regular and nondefective for a robust solution such that the assigned closed loop poles are insensitive to perturbations in the system data the same conditions must hold it can be shown that these conditions are also sufficient for the existence of a feedback which assigns the maximum possible number of finite poles with regularity these results provide the basis of a procedure for constructing closed loop semistate systems with given poles guaranteed regularity and maximum robustness

newton type minimization via the lanczos method this paper discusses the use of the linear conjugate gradient method developed via the lanczos method in the solution of large scale unconstrained minimization problems at each iteration of a newton type method the direction of search is defined as the solution of a quadratic subproblem when the number of variables is very large this subproblem may be solved using the linear conjugate gradient method of hestenes and stiefel we show how the equivalent lanczos characterization of the linear conjugate gradient method may be exploited to define a modified newton method which can be applied to problems that do not necessarily have positive definite hessian matrices at all points of the region of interest this derivation also makes it possible to compute a negative curvature direction at a stationary point the idea of a truncated newton method is to perform only a limited number of iterations of the quadratic subproblem this effectively gives a search direction that interpolates between the steepest de

solvability conditions consistency and weak consistency for linear differential algebraic equations and time invariant singular systems the general case we present several solvability concepts for linear differential algebraic equations daes with constant coefficients on the positive time axis as well as for the associated singular systems and investigate under which conditions these concepts are met next we derive necessary and sufficient conditions for global consistency of initial conditions for the dae as well as for the system and generalize these conditions with respect to our concept of weak consistency our distributional approach enables us to generalize results in an earlier paper where singular systems are assumed to have a regular pencil in the sense of gantmacher in particular we establish that global weak consistency in the system sense is equivalent to impulse controllability

an arithmetic for matrix pencils theory and new algorithms this paper introduces arithmetic like operations on matrix pencils the pencil arithmetic operations extend elementary formulas for sums and products of rational numbers and include the algebra of linear transformations as a special case these operations give an unusual perspective on a variety of pencil related computations we derive generalizations of monodromy matrices and the matrix exponential a new algorithm for computing a pencil arithmetic generalization of the matrix sign function does not use matrix inverses and gives an empirically forward numerically stable algorithm for extracting deflating subspaces

euler s constant euler s work and modern developments this paper has two parts the first part surveys euler s work on the constant 0 57721 bearing his name together with some of his related work on the gamma function values of the zeta function and divergent series the second part describes various mathematical developments involving euler s constant as well as another constant the euler gompertz constant these developments include connections with arithmetic functions and the riemann hypothesis and with sieve methods random permutations and random matrix products it also includes recent results on diophantine approximation and transcendence related to euler s constant

asymptotic estimates for stieltjes constants a probabilistic approach let n n 0 be the sequence of stieltjes constants appearing in the laurent expansion of the riemann zeta function we obtain explicit upper bounds for n whose order of magnitude is as n tends to

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

descriptor systems without controllability at infinity this paper concerns the structure that can be achieved by feedback in descriptor systems that lack controllability at infinity staircase and double staircase condensed forms obtained through a sequence of orthogonal state transformations display when and how feedback can be used to achieve minimal index furthermore they reveal that the modes that are uncontrollable at infinity have a fixed minimal index that cannot be reduced by feedback however this fixed higher index part of the control system is constrained to be zero in an appropriate coordinate system provided the initial conditions are consistent the remainder is a reduced order system that is controllable at infinity that can be made to have index one by feedback

on reducing infinite eigenvalues of regular pencils by a nonequivalence transformation abstract we develop an algorithm for the deflation of the invariant subspace corresponding to infinite eigenvalues of a regular pencil by a nonequivalence transformation this method is a generalization of wielandt s algorithm for the deflation of the invariant subspace of a constant matrix

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

ab algorithm and its modifications for the spectral problems of linear pencils of matrices this paper describes and algorithm and its modifications for solving spectral problems for linear pencils of matrices both regular as well as singular

root polynomials and their role in the theory of matrix polynomials abstract we develop a complete and rigorous theory of root polynomials of arbitrary matrix polynomials i e either regular or singular and study how these vector polynomials are related to the spectral properties of matrix polynomials we pay particular attention to the so called maximal sets of root polynomials and prove that they carry complete information about the eigenvalues finite or infinite of matrix polynomials and that they are related to the matrices that transform any matrix polynomial into its smith form in addition we describe clearly for the first time in the literature the extremality properties of such maximal sets and identify some of them whose vectors have minimal grade once the main theoretical properties of root polynomials have been established the interaction of root polynomials with three problems that have attracted considerable attention in the literature is analyzed more precisely we study the change of root polynomials under rational transformations or reparametrizations of matrix polynomials the recovery of the root polynomials of a matrix polynomial from those of its most important linearizations and the relationship between the root polynomials of two dual pencils we emphasize that for the case of regular matrix polynomials all the results in this paper can be translated into the language of jordan chains as a consequence of the well known relationship between root polynomials and jordan chains therefore a number of open problems are also solved for jordan chains of regular matrix polynomials we also briefly discuss how root polynomials can be used to define eigenvectors and eigenspaces for singular matrix polynomials

on the zeros of the riemann zeta function in the critical strip iv etude de la localisation des zeros complexes de la fonction zeta de riemann verification numerique de l hypothese de riemann pour les 1 500 000 001 premiers zeros de la forme it dans la bande critique avec 0 t 545 439 823 215 tous ces zeros ayant pour partie reelle 1 2

duality of matrix pencils wong chains and linearizations we consider two theoretical tools that have been introduced decades ago but whose usage is not widespread in modern literature on matrix pencils one is dual pencils a pair of pencils with the same regular part and related singular structures they were introduced by v kublanovskaya in the 1980s the other is wong chains families of subspaces associated with possibly singular matrix pencils that generalize jordan chains they were introduced by k t wong in the 1970s together dual pencils and wong chains form a powerful theoretical framework to treat elegantly singular pencils in applications especially in the context of linearizations of matrix polynomials we first give a self contained introduction to these two concepts using modern language and extending them to a more general form we describe the relation between them and show how they act on the kronecker form of a pencil and on spectral and singular structures eigenvalues eigenvectors and minimal bases then we present several new applications of these results to more recent topics in matrix pencil theory including constraints on the minimal indices of singular hamiltonian and symplectic pencils new sufficient conditions under which pencils in l1 l2 linearization spaces are strong linearizations a new perspective on fiedler pencils and a link between the moller stetter theorem and some linearizations of matrix polynomials

new summation relations for the stieltjes constants the stieltjes constants have been of interest for over a century yet their detailed behaviour remains under investigation these constants appear in the laurent expansion of the hurwitz zeta funct

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

hyperelliptic jacobians and isogenies abstract in this note we mainly consider abelian varieties isogenous to hyperelliptic jacobians in the first part we prove that a very general hyperelliptic jacobian of genus g 4 is not isogenous to a non hyperelliptic jacobian as a consequence we obtain that the intermediate jacobian of a very general cubic threefold is not isogenous to a jacobian another corollary tells that the jacobian of a very general d gonal curve of genus g 4 is not isogenous to a different jacobian in the second part we consider a closed subvariety y a g of the moduli space of principally polarized varieties of dimension g 3 we show that if a very general element of y is dominated by the jacobian of a curve c and dim y 2 g then c is not hyperelliptic in particular if the general element in y is simple its kummer variety does not contain rational curves finally we show that a closed subvariety y m g of dimension 2 g 1 such that the jacobian of a very general element of y is dominated by a hyperelliptic jacobian is contained either in the hyperelliptic or in the trigonal locus

differentiation evens out zero spacings if f is a polynomial with all of its roots on the real line then the roots of the derivative f are more evenly spaced than the roots of f the same holds for a real entire function of order 1 with all its zeros on a line in particular we show that if f is entire of order 1 and has sufficient regularity in its zero spacing then under repeated differentiation the function approaches after normalization the cosine function we also study polynomials with all their zeros on a circle and we find a close analogy between the two situations this sheds light on the spacing between zeros of the riemann zeta function and its connection to random matrix polynomials

an asymptotic form for the stieltjes constants gamma k a and for a sum s gamma n appearing under the li criterion we present several asymptotic analyses for quantities associated with the riemann and hurwitz zeta functions we first determine the leading asymptotic behavior of the stieltjes constants k a these constants appear in the regular part of the laurent expansion of the hurwitz zeta function we then use asymptotic results for the laguerre polynomials l n to investigate a certain sum s n involving the constants k 1 that appears in application of the li criterion for the riemann hypothesis we confirm the sublinear growth of s n n which is consistent with the validity of the riemann hypothesis

the distribution of the summatory function of the mobius function the summatory function of the mobius function is denoted m x in this article we deduce conditional results concerning m x assuming the riemann hypothesis and a conjecture of gonek and hejhal on the negative moments of the riemann zeta function assuming these conjectures we show that m x when appropriately normalized possesses a limiting distribution and also that a strong form of the weak mertens conjecture is true finally we speculate on the lower order of m x by studying the constructed distribution function

a generalized state space for singular systems systems of the form e dot x ax bu y cx with e singular are studied of particular interest are the impulsive modes that may appear in the free response of such systems when arbitrary initial conditions are permitted modes that are associated with natural system frequencies at infinity a generalized definition of system order that incorporates these impulsive degrees of freedom is proposed and concepts of controllability and observability are defined for the impulsive modes allowable equivalence transformations of such singular systems are specified the present framework is shown to overcome several difficulties inherent in other treatments of singular systems and to extend in a natural and satisfying way many results previously known only for regular state space systems

estimates of generalized stieltjes constants with a quasi geometric rate of decay we approximate each generalized stieltjes constant n a by means of a finite sum involving bernoulli numbers such an approximation has a quasi geometric rate of convergence which improves as re a increases a more detailed analysis including numerical computations is carried out for the constants 1 and 1 1 the key point in the proof is a probabilistic representation of the aforementioned constants obtained as a consequence of a differential calculus concerning the gamma process

relations and positivity results for the derivatives of the riemann ξ function we present and evaluate the integer order derivatives of the riemann xi function these derivatives contain logarithmic integrals of powers multiplying a specific jacobi theta function and as such can be alternatively viewed as certain mellin transforms at integer argument we describe how the derivatives at s 0 s and s 1 can be evaluated exactly we further show based upon a novel representation that the even order derivatives at s are all positive as are all derivatives at s 1 an expression is presented for the derivatives on the critical line which may be useful in studying the zeros of the function t it

asymptotic estimation of ξ 2n 1 2 on a conjecture of farmer and rhoades we verify a very recent conjecture of farmer and rhoades on the asymptotic rate of growth of the derivatives of the riemann xi function at s 1 2 we give two separate proofs of this result with the more general method not restricted to s 1 2 we briefly describe other approaches to our results give a heuristic argument and mention supporting numerical evidence

the stieltjes function definition and properties definition et proprietes d une fonction liee au developpement en serie de laurent de la fonction zeta de riemann au voisinage du point singulier s 1 fonction de stieltjes

zeta values and differential operators on the circle abstract the fock representation of the virasoro lie algebra is extended to a larger graded lie subalgebra of the algebra of differential operators on the circle the central cocycle is related to values of the riemann zeta function at odd negative integers the corresponding generating function is related to eisenstein series for sl 2 z an analogous result is obtained for values of l series associated to even dirichlet characters using differential operators of infinite order

sieve methods for odd perfect numbers using a new factor chain argument we show that 5 does not divide an odd perfect number indivisible by a sixth power applying sieve techniques we also find an upper bound on the smallest prime divisor putting this together we prove that an odd perfect number must be divisible by the sixth power of a prime or its smallest prime factor lies in the range 10 8 p 10 1000 these results are generalized to much broader situations

the hexagonal versus the square lattice schmutz schaller s conjecture regarding the lengths of the hexagonal versus the lengths of the square lattice is shown to be true the proof makes use of results from computational prime number theory using an identity due to selberg it is shown that in principle the conjecture can be resolved without using computational prime number theory by our approach however this would require a huge amount of computation

new results concerning power series expansions of the riemann xi function and the li keiper constants the riemann hypothesis is equivalent to the li criterion governing a sequence of real constants that are certain logarithmic derivatives of the riemann xi function evaluated at unity a new represe

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

on some series representations of the hurwitz zeta function a variety of infinite series representations for the hurwitz zeta function are obtained particular cases recover known results while others are new specialization of the series representations apply to the riemann zeta function leading to additional results the method is briefly extended to the lerch zeta function most of the series representations exhibit fast convergence making them attractive for the computation of special functions and fundamental constants

positive solutions to x a bx 1 b abstract we study the positive semidefinite solutions to the matrix equation x a bx 1 b under the assumption that a 0 we show that a positive solution exists if and only if a certain block tridiagonal operator is positive in which case the solution is given by the shorted operator or generalized schur complement of that operator

parallel algorithm for solving some spectral problems of linear algebra abstract an algorithm is described for computing the deflating subspaces of a regular linear matrix pencil b a more precisely the algorithm is intended to compute the projection matrices p and i p onto the deflating subspaces of matrix pencils corresponding to the eigenvalues inside and outside the unit circle this algorithm can be considered as an orthogonal version of the matrix sign function method it possesses a quadratic convergence rate and simultaneously with the projection operators operators computes a condition number of the problem we point out the extent to which this algorithm can be parallelized applications are made to solving the riccati equation

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

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

on sums of seven cubes we show that every integer between 1290741 and 3 375 10 12 is a sum of 5 non negative cubes from which we deduce that every integer which is a cubic residue modulo 9 and an invertible cubic residue modulo 37 is a sum of 7 non negative cubes

necessary and sufficient conditions for the existence of a positive definite solution of the matrix equation x a x 1a q abstract we consider the problem of when the matrix equation x a x 1 a q has a positive definite solution here q is positive definite we study both the real and the complex case this equation plays a crucial role in solving a special case of the discrete time riccati equation we present both necessary and sufficient conditions for its solvability this result is obtained by using an analytic factorization approach moreover we present algebraic recursive algorithms to compute the largest and smallest the solution of the equation respectively finally we discuss the number of solutions

a new look at pencils of matrix valued functions abstract matrix pencils depending on a parameter and their canonical forms under equivalence are discussed the study of matrix pencils or generalized eigenvalue problems is often motivated by applications from linear differential algebraic equations daes based on the weierstrass kronecker canonical form of the underlying matrix pencil one gets existence and uniqueness results for linear constant coefficients daes in order to study the solution behavior of linear daes with variable coefficients one has to look at new types of equivalence transformations this then leads to new canonical forms and new invariances for pencils of matrix valued functions we give a survey of recent results for square pencils and extend these results to nonsquare pencils furthermore we partially extend the results for canonical forms of hermitian pencils and give new canonicalforms there too based on these results we obtain new existence and uniqueness theorems for differential algebraic systems which generalize the classical results of weierstrass and kronecker

regularization of linear descriptor systems with variable coefficients we study linear descriptor control systems with rectangular variable coefficient matrices we introduce condensed forms for such systems under equivalence transformations and use these forms to detect whether the system can be transformed to a uniquely solvable closed loop system via state or derivative feedback we show that under some mild assumptions every such system consists of an underlying square subsystem that behaves essentially like a standard state space system plus some solution components that are constrained to be zero

an effective asymptotic formula for the stieltjes constants the stieltjes constants k appear in the coefficients in the regular part of the laurent expansion of the riemann zeta function s about its only pole at s 1 we present an asymptotic expression for k for k 1 this form encapsulates both the leading rate of growth and the oscillations with k furthermore our result is effective for computation consistently in close agreement for both magnitude and sign for even moderate values of k comparison to some earlier work is made

chebyshev s bias for composite numbers with restricted prime divisors let x d a denote the number of primes p x with p a modd chebyshev s bias is the phenomenon for which more often x d n x d r than the other way around where n is a quadratic nonresidue mod d and r is a quadratic residue mod d if x d n x d r for every x up to some large number then one expects that n x d n n x d r for every x here n x d a denotes the number of integers n x such that every prime divisor p of n satisfies p a mod d in this paper we develop some tools to deal with this type of problem and apply them to show that for example n x 4 3 n x 4 1 for every x in the process we express the so called second order landau ramanujan constant as an infinite series and show that the same type of formula holds for a much larger class of constants

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

an efficient algorithm for the hurwitz zeta and related functions a simple class of algorithms for the efficient computation of the hurwitz zeta and related special functions is given the algorithms also provide a means of computing fundamental mathematical constants to arbitrary precision a number of extensions as well as numerical examples are briefly described the algorithms are easy to implement and compete with euler maclaurin summation based methods

generalized euler constants for arithmetical progressions the work of lehmer and briggs on euler constants in arithmetical progressions is extended to the generalized euler constants that arise in the laurent expansion of g s about s 1 the results are applied to the summation of several classes of slowly converging series a table of the constants is provided

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

a quadratically convergent qr like method without shifts for the hermitian eigenvalue problem abstract we propose a new qr like algorithm symmetric squared qr ssqr method that can be readily parallelized using commonly available parallel computational primitives such as matrix matrix multiplication and qr decomposition the algorithm converges quadratically and the quadratic convergence is achieved through a squaring technique without utilizing any kind of shifts we provide a rigorous convergence analysis of ssqr and derive structures for several of the important quantities generated by the algorithm we also discuss various practical implementation issues such as stopping criteria and deflation techniques we demonstrate the convergence behavior of ssqr using several numerical examples

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