on the zeros of the ramanujan t dirichlet series in the critical strip we describe computations which show that each of the first 12069 zeros of the ramanujan r dirichlet series of the form it in the region 0 t 6397 is simple and lies on the line 6 the failures of gram s law in this region are also noted the first 5018 zeros and 2228 successive zeros beginning with the 20001st zero are also calculated the distribution of the normalized spacing of the zeros is examined and it appears to be that of the eigenvalues of random matrices these comptuations are done with a berry keating formula for the r dirichlet series and evaluated using mathematica

convergence rates in stochastic adaptive tracking abstract for stochastic control systems described by the armax model with unknown matrix coefficients the stochastic adaptive control is designed so that the parameter estimates converge to the true values with a rate of convergence o log n log log n c n a with a 0 c 1 and the tracking error tends to its minimum value at a speed of o n 1 2e with e 0

discrete time stochastic adaptive control this paper establishes global convergence of a stochastic adaptive control algorithm for discrete time linear systems it is shown that with probability one the algorithm will ensure the system inputs and outputs are sample mean square bounded and the conditional mean square output tracking error achieves its global minimum possible value for linear feedback control thus asymptotically the adaptive control algorithm achieves the same performance as could be achieved if the system parameters were known

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

the sum of like powers of the zeros of the riemann zeta function on analyse une methode de calcul de la somme r p r ou r est un entier superieur a 1 la somme etant prise sur tous les zeros complexes de la fonction zeta de riemann

adaptive control with recursive identification for stochastic linear systems this paper presents a stochastic adaptive control algorithm which is shown to possess the following properties when applied to a possibly unstable inverse stable linear stochastic system with unknown parameters whenever that system satisfies a certain positive real condition on its moving average noise dynamics 1 the adaptive control part of the algorithm stabilizes and asymptotically optimizes the behavior of the system in the sense that the limit of the sample mean square variation of the output around a given demand level equals that of a minimum variance control strategy implemented with known parameters this optimal behavior is subject to an offset 2where 2is the variance of a dither signal added to the control action in order to produce a continually disturbed control for mu 2 0 it is shown that the input output process satisfies a persistent excitation property and hence subject to a simple identifiability condition the next property holds 2 the observed input and output of the controlled system may be taken as inputs to an approximate maximum likelihood algorithm aml which generates strongly consistent estimates of the system s parameters results are presented for the scalar and multivariable cases

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

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

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

stochastic adaptive pole zero assignment with convergence analysis abstract the adaptive control u n is designed for the stochastic system a z y n 1 b z u n c z w n 1 with unknown constant matrix coefficients in the polynomials a z b z and c z in the shift back operator with the purposes that 1 the unknown matrices are strongly consistently estimated and 2 the poles and zeros are replaced in such a way that the system itself is transferred to a 0 z y n 1 b 0 z u n 0 e n 1 with given a 0 z b 0 z and u n 0 so that the pole zero assignment error e n 1 is minimized the problem of adaptive pole zero assignment combined with tracking is also considered in this paper conditions used are imposed only on a z b z and c z

quasi least squares identification and its strong consistency simple identification algorithms are proposed for both continuous and discrete time systems with correlated noise by a combined method connecting the martingale method with the ordinary differential equation method quite simple conditions are obtained to guarantee strong consistency of the proposed algorithms which give estimates for the unknown coefficients in both the dynamic system and the noise model

on certain sums over the non trivial zeta zeroes let z denote the riemann zeta function and x s s 2 s 1 p s 2 g s 2 z s the classical completed zeta function where g is the gamma function riemann 1859 1860 edwards 1974 ivic 1985 titchmarsh 1986 within the critical strip s 0 re s 1 the complex zeroes of z and x coincide and we denote them by r the x function is entire of order 1 and of maximal type herein we mainly investigate certain sums bn over complex zeta function zeroes we provide various representations and properties of these sums with the main specific results stated in 2 we also supply some remarks on the li criterion li 1997 for the riemann hypothesis rh we recall that the li equivalence for the rh results as a necessary and sufficient condition that the logarithmic derivative of the function x 1 1 z be analytic in the unit disc this obtains from a conformal map of the critical strip to this disc the equivalence li 1997 states that a necessary and sufficient condition for the non trivial zeroes of the z function to lie on the critical line re s 1 is that constants lk 1 are non negative for every integer k the sequence ln 1

the strong consistency of the stochastic gradient algorithm of adaptive control by use of a technique due to chen sufficient conditions are established for the strong consistency of the stochastic gradient sg algorithm for mimo armax stochastic systems without monitoring this result is then used in conjunction with the method of disturbed adaptive controls introduced in 1 2 hence it is shown that the sg algorithm generates strongly consistent parameter estimates while it is operating as a part of the sg algorithm of adaptive control of goodwin ramadge and caines

on the laurent series for the epstein zeta function the epstein zeta function d m s n z d n m n t s 2 where is a real symmetric and invertible d d matrix and is a d dimensional row vector with integer coordinates is considered the prime on the sum indicates that the term should be excluded it is known that has a laurent series expansion about the singular point s d which can be written in the form d m s 1 a m d s d in this paper we shall show that the coefficient can be accurately calculated using rapidly convergent series which involve the meijer g function exact formulae are also derived for when m u n 1 0 0 n with the results for are then used to establish several mathematical identities involving summations of generalized stieltjes constants next the laurent series for where is the d d unit matrix is briefly discussed for the cases and 8 finally a new application of the results in lattice statistics is described

interface pinning and slow ordering kinetics on infinitely ramified fractal structures we investigate the time dependent ginzburg landau tdgl equation for a non conserved order parameter on an infinitely ramified deterministic fractal lattice employing two alternative methods the auxiliary field approach and a numerical method of integration of the equations of evolution in the first case the domain size evolves with time as l t sim t 1 d w where d w is the anomalous random walk exponent associated with the fractal and differs from the normal value 2 which characterizes all euclidean lattices such a power law growth is identical to the one observed in the study of the spherical model on the same lattice but fails to describe the asymptotic behavior of the numerical solutions of the tdgl equation for a scalar order parameter in fact the simulations performed on a two dimensional sierpinski carpet indicate that after an initial stage dominated by a curvature reduction mechanism a la allen cahn the system enters in a regime where the domain walls between competing phases are pinned by lattice defects the lack of translational invariance determines a rough free energy landscape the existence of many metastable minima and the suppression of the marginally stable modes which in translationally invariant systems lead to power law growth and self similar patterns on fractal structures as the temperature vanishes the evolution is frozen since only thermally activated processes can sustain the growth of pinned domains

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

summation properties of the η j and li constants we find new summatory and other properties of the constants h j entering the laurent expansion of the logarithmic derivative of the riemann zeta function about s 1 we relate these constants to other coefficients and functions appearing in the theory of the zeta function in particular connections to the li equivalence of the riemann hypothesis are discussed and quantitatively developed the validity of the riemann hypothesis is reduced to the condition of the sublinear order of a certain alternating binomial sum

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

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

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

front type solutions of fractional allen cahn equation abstract super diffusive front dynamics have been analysed via a fractional analogue of the allen cahn equation one dimensional kink shape and such characteristics as slope at origin and domain wall dynamics have been computed numerically and satisfactorily approximated by variational techniques for a set of anomaly exponents 1 2 the dynamics of a two dimensional curved front has been considered also the time dependence of coarsening rates during the various evolution stages was analysed in one and two spatial dimensions

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

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

computation and theory of extended mordell tornheim witten sums we consider some fundamental generalized mordell tornheim witten mtw zeta function values along with their derivatives and explore connections with multiple zeta values mzvs to achieve this we make use of symbolic integration high precision numerical integration and some interesting combinatorics and special function theory our original motivation was to represent unresolved constructs such as eulerian log gamma integrals we are able to resolve all such integrals in terms of a mtw basis we also present for a substantial subset of mtw values explicit closed form expressions in the process we signicantly extend methods for high precision numerical computation of polylogarithms and their derivatives with respect to order

persistence of excitation in extended least squares in least squares parameter estimation schemes persistency of excitation conditions on the plant states are required for consistent estimation in the case of extended least squares the persistency conditions are on the state estimates here these persistency of excitation conditions are translated into sufficiently rich conditions on the plant noise and inputs in the case of adaptive minimum variance control schemes the sufficiently rich conditions are on the noise and specified output trajectory with sufficiently rich input signals guaranteed convergence rates of prediction errors improve and it is conjectured that the algorithms are consequently more robust

strong consistency of parameter estimates in direct self tuning control algorithms based on stochastic approximation in this paper convergence of the regulator parameter estimates generated by direct self tuning control algorithms of stochastic approximation type is considered the process is supposed to be represented by an armax model with an arbitrary delay while a dither signal converging to zero in the mean square sense is superimposed on a deterministic reference the strong consistency of the regulator parameter estimates is proved under assumptions concerning the estimator complexity and the properties of the optimal regulator transfer function for a broad class of dither signals the impact of a priori information on the dimension of the estimator and the consistency of the parameter estimates is indicated it is demonstrated that the classical persistence of excitation condition is not satisfied for the analyzed algorithms

convergence rate of least squares identification and adaptive control for stochastic systems the strong consistency and the convergence rate of least squares identification for the multidimensional armax model are established under some decaying excitation conditions which are satisfied if both input and output do not grow too fast and the attenuating excitation technique is applied the parameter identification results are applied to adaptive control systems with a quadratic loss function the rate of convergence of the loss function to its minimum is also obtained

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

entropic measure of spatial disorder for systems of finite sized objects abstract we consider the relative configurational entropy per cell s as a measure of the degree of spatial disorder for systems of finite sized objects it is highly sensitive to deviations from the most spatially ordered reference configuration of the objects when applied to a given binary image it provides the quantitatively correct results in comparison to its point object version on examples of simple cluster configurations two dimensional sierpinski carpets and population of interacting particles the behaviour of s is compared with the normalized information entropy h introduced by van siclen phys rev e 56 1997 5211 for the latter example the additional middle scale features revealed by our measure may indicate for the traces of self similar structure of the weakly ramified clusters in the thermodynamic limit the formula for s is also given

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

convergence of a multivariable self tuning controller with integral action abstract in this note global convergence is established for the multivariable self tuning control algorithm presented in 11 the plant is assumed to be described by a carima model the control algorithm relies on the generalized minimum variance approach the convergence proof is carried out by resorting to an interlaced stochastic gradient identification algorithm

formulas for higher derivatives of the riemann zeta function the functional equation for c s is used to obtain formulas for all derivatives t k s a closed form evaluation of t k 0 is given and numerical values are computed to

improved results on the mertens conjecture we study the mertens conjecture and revisit and improve the original constructivedisproof of janos pintz we obtain a new lower bound for the minimal counterexample and new numerical results for this conjecture

recursive system identification and adaptive control by use of the modified least squares algorithm in this paper we first use a weaker than persistent excitation condition to establish the strong consistency of mls for mimo stochastic systems without monitoring and give a comparison between various conditions for consistency then we show the global convergence for adaptive tracking based on mls and finally the adaptive tracking and strong consistency results are combined

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