zeros of l functions attached to maass forms this paper considers psl 2 automorphic cusp forms the maass forms and their associated dirichlet series there are two major results the first uses real variable techniques to give an explicit reconstruction of the cusp form in terms of the dirichlet series this representation is valid in the entire upper half plane the second uses this representation to show that the dirichlet series has many zeros on its critical line this is the analogue of the hardy littlewood result for the riemann zeta function

improved derivative bounds of the rational quadratic bezier curves a new derivative bound of the rational quadratic bezier curves is obtained it is an improvement over the bound which was regarded as a sharp bound in a recent paper also the new bound is superior to the existing bounds given by some other authors

selberg s orthonormality conjecture and joint universality of l functions in the paper we introduce a new method how to use only an orthonormality relation of coefficients of dirichlet series defining given l functions from the selberg class to prove joint universality

joint universality theorems for pairs of automorphic zeta functions in this paper we establish the joint universality theorems for pairs consisting of the riemann zeta function automorphic l functions associated with holomorphic hecke eigen cusp forms rankin selberg l functions and symmetric square l functions

unique factorization results for semigroups of l functions in this paper we prove the unique factorization of certain subsemigroups of the extended selberg class of l functions these semigroups are generated by the functions of degree 0 and 1 we also consider extensions of such results and applications to classical l functions

a remark on the uniqueness of the dirichlet series with a riemann type function equation abstract we show that if for a nonzero complex number c the inverse images l 1 1 c and l 2 1 c of two functions in the extended selberg class are the same then l 1 s and l 2 s must be identical

why saturated probability spaces are necessary abstract an atomless probability space a p is said to have the saturation property for a probability measure on a product of polish spaces x y if for every random element f of x whose law is marg x there is a random element g of y such that the law of f g is a p is said to be saturated if it has the saturation property for every such we show each of a number of desirable properties holds for every saturated probability space and fails for every non saturated probability space these include distributional properties of correspondences such as convexity closedness compactness and preservation of upper semi continuity and the existence of pure strategy equilibria in games with many players we also show that any probability space which has the saturation property for just one good enough measure or which satisfies just one good enough instance of the desirable properties must already be saturated our underlying themes are 1 there are many desirable properties that hold for all saturated probability spaces but fail everywhere else 2 any probability space that out performs the lebesgue unit interval in almost any way at all is already saturated

improved bounds on the magnitude of the derivative of rational bezier curves abstract in this paper we derive some new derivative bounds of rational bezier curves according to some existing identities and inequalities the comparison of the new bounds with some existing ones is also presented

on the mod gaussian convergence of a sum over primes we prove mod gaussian convergence for a dirichlet polynomial which approximates mathrm im log zeta 1 2 it this dirichlet polynomial is sufficiently long to deduce selberg s central limit theorem with an explicit error term moreover assuming the riemann hypothesis we apply the theory of the riemann zeta function to extend this mod gaussian convergence to the complex plane from this we obtain that mathrm im log zeta 1 2 it satisfies a large deviation principle on the critical line results about the moments of the riemann zeta function follow

a hybrid euler hadamard product for the riemann zeta function we use a smoothed version of the explicit formula to find an accurate pointwise approximation to the riemann zeta function as a product over its nontrivial zeros multiplied by a product over the primes we model the first product by characteristic polynomials of random matrices this provides a statistical model of the zeta function which involves the primes in a natural way we then employ the model in a heuristic calculation of the moments of the modulus of the zeta function on the critical line for the second and fourth moments we establish all of the steps in our approach rigorously this calculation illuminates recent conjectures for these moments based on connections with random matrix theory

on common zeros of l functions we will prove that two l functions l1 l2 satisfying the same functional equation must be identically equal if sufficiently many zeros of l1 are also zeros of l2

on the structure of the selberg class i 0 d 1 the selberg class s is a rather general class of dirichlet series with functional equation and euler product and can be regarded as an axiomatic model for the global l functions arising from number theory and automorphic representations one of the main problems of the selberg class theory is to classify the elements of s such a classication is based on a real valued invariant d called degree and the degree conjecture asserts that d 2 n for every l function in s the degree conjecture has been proved for d 5 3 and in this paper we extend its validity to d 2 the proof requires several new ingredients in particular a rather precise description of the properties of certain nonlinear twists associated with the l functions in s

functional independence of the singularities of a class of dirichlet series we deal with the algebraic independence and more generally with the functional independence of the singularities of log f j s j 1 n and of f j f j s j 1 n where f j s are functions in the selberg class in particular we prove the following results i if log f 1 s log f n s are linearly independent over inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 01i then p log f 1 s log f n s s has infinitely many singularities in the half plane provided p inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 02i x 1 x n 1 with deg p 0 as a polynomial in the first n variables and ii if p inline graphic xmlns xlink http www w3 org 1999 xlink xlink href 03i x 1 x n with deg p 0 then p f 1 f 1 s f n f n s is either constant or has infinitely many singularities in the half plane 0

l functions as distributions we define an axiomatic class of l functions extending the selberg class we show in particular that one can recast the traditional conditions of an euler product analytic continuation and functional equation in terms of distributional identities akin to weil s explicit formula the generality of our approach enables some new applications for instance we show that the l function of any cuspidal automorphic representation of mathrm gl 3 mathbb a mathbb q has infinitely many zeros of odd order

on the bounds of the derivative of rational bezier curves according to some identities and inequalities about the intermediate weights and control points of the de casteljau algorithm for rational bezier curve we derive a new bound of the derivative of rational bezier curves the comparison of the new bounds with some existing ones is also presented based on some previous bounds the bound presented in this paper and many numerical examples we propose a conjecture of the bound for the derivative of rational bezier curves

gaps between zeros of gl 2 l functions abstract let l s f be an l function associated to a primitive holomorphic or maass cusp form f on gl 2 over q combining mean value estimates of montgomery and vaughan with a method of ramachandra we prove a formula for the mixed second moments of derivatives of l 1 2 i t f and via a method of hall use it to show that there are infinitely many gaps between consecutive zeros of l s f along the critical line that are at least 3 1 732 times the average spacing using general pair correlation results due to murty and perelli in conjunction with a technique of montgomery we also prove the existence of small gaps between zeros of any primitive l function of the selberg class in particular when f is a primitive holomorphic cusp form on gl 2 over q we prove that there are infinitely many gaps between consecutive zeros of l s f along the critical line that are at most 0 823 times the average spacing

a convexity result for the range of vector measures with applications to large economies on a boolean algebra we consider the topology u induced by a finitely additive measure mu with values in a locally convex space and formulate a condition on u that is sufficient to guarantee the convexity and weak compactness of the range of mu this result a la lyapunov extends those obtained in khan sagara 2013 to the finitely additive setting through a more direct and less involved proof we will then give an economical interpretation of the topology u in the framework of coalitional large economies to tackle the problem of measuring the bargaining power of coalitions when the commodity space is infinite dimensional and locally convex we will show that our condition on u plays the role of the many more agents than commodities condition introduced by rustichini and yannelis in 1991 as a consequence of the convexity theorem we will obtain two straight generalizations of schmeidler s and vind s theorems on the veto power of coalitions of arbitrary economic weight

on the location of the zero free half plane of a random epstein zeta function in this note we study for a random lattice l of large dimension n the supremum of the real parts of the zeros of the epstein zeta function e n l s and prove that this random variable scaled by n 1 has a limit distribution which we give explicitly this limit distribution is studied in some detail in particular we give an explicit formula for its distribution function furthermore we obtain a limit distribution for the frequency of zeros of e n l s in vertical strips contained in the half plane mathfrak r s frac n 2

derivative bound estimations on rational conic bezier curves abstract in this paper we derive a bound on derivatives of rational conic bezier curves both theoretical analysis and numerical examples show that our bound is sharper than some existed ones discussions on optimal bounds for the derivatives of rational conic bezier curves are also provided

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

new minimal bounds for the derivatives of rational bezier paths and rational rectangular bezier surfaces new minimal bounds are derived for the magnitudes of the derivatives of the rational bezier paths and the rational rectangular bezier surface patches of arbitrary degree which improve previous work of this type in many cases moreover our new bounds are explicitly given by simple and closed form expressions an important advantage of the closed form expressions is that they allow us to prove that our bounds are sharp under certain well defined conditions some numerical examples highlighting the potential of the new bounds in providing improved estimates are given in an appendix

some remarks on the convergence of the dirichlet series of l functions and related questions first we show that the abscissae of uniform and absolute convergence of dirichlet series coincide in the case of l functions from the selberg class mathcal s we also study the latter abscissa inside the extended selberg class indicating a different behavior in the two classes next we address two questions about majorants of functions in mathcal s showing links with the distribution of the zeros and with independence results

on the mean values of dirichlet l functions we study the 2kth power moment of dirichlet l functions l s at the centre of the critical strip s 2 where the average is over all primitive characters mod q we extend to this case the hybrid euler hadamard product results of gonek hughes and keating for the riemann zeta function this allows us to recover conjectures for the moments based on random matrix models incorporating the arithmetical terms in a natural way