Region Of Absolute Convergence Research Articles

Analytic properties of Ohno function

Ohno's relation is a well-known relation on the field of the multiple zeta values and has an interpolation to complex function. In this paper, we call its complex function Ohno function and study it. We consider the region of absolute convergence, give some new expressions, and show new relations of the function. We also give a direct proof of the interpolation of Ohno's relation.

ДЗЕТА-ФУНКЦИЯ МОНОИДОВ НАТУРАЛЬНЫХ ЧИСЕЛ С ОДНОЗНАЧНЫМ РАЗЛОЖЕНИЕМ НА ПРОСТЫЕ МНОЖИТЕЛИ1

In this paper we consider a new class of Dirichlet series, the zeta functions of monoids of natural numbers. The inverse Dirichlet series for the zeta function of monoids of natural numbers are studied. It is shown that the existence of an Euler product for the zeta function of a monoid is related to the uniqueness of the factorization into prime factors in this monoid. The notion of coprime sets of natural numbers is introduced and it is shown that for such sets the multiplicativity of minimal monoids and corresponding zeta-functions of monoids takes place. It is shown that if all prime elements of a monoid are prime numbers, then the characteristic function of the monoid is a multiplicative function and in this case the zeta function of the monoid is a generalized L-function. Various examples of monoids and corresponding zeta functions of monoids are considered. The relation between the inversion of the zeta function of a monoid and the generalized Mo¨bius function on a monoid as a partially ordered set is studied by means of the divisibility of natural numbers. A number of properties of the zeta functions of monoids of natural numbers with a unique decomposition into prime factors are obtained. The paper deals with taking the logarithm of an Eulerian product as a function of a complex argument. It is shown that a continuous function that determines the value of the logarithm of an Euler product runs through all branches of the infinite-valued function of the logarithm near its pole. The corollaries on the value of a complex-valued function of a special form near a singular point are obtained. These properties imply statements about the values of the Riemann zeta function near the boundary of the region of absolute convergence. Using Bertrand’s postulate, infinite exponential sequences of prime numbers are introduced. It is shown that corresponding zeta-functions of monoids of natural numbers converge absolutely in the whole half-plane with a positive real part. Since such zeta-functions of monoids of natural numbers can be decomposed into an Euler product in the whole region of absolute convergence, they do not have zeros in the entire half-plane with a positive real part. In conclusion, topical problems with zeta-functions of monoids of natural numbers that require further investigation are considered.

Hurwitz–Lerch zeta and Hurwitz–Lerch type of Euler–Zagier double zeta distributions

In this paper, we give Hurwitz–Lerch zeta distributions with [Formula: see text] by using the Gamma function. Moreover, we define Hurwitz–Lerch type of Euler–Zagier double zeta distributions not only in the region of absolute convergence but also outside the region of absolute convergence.

Zeros ofL-functions outside the critical strip

For a wide class of Dirichlet series associated to automorphic forms, we show that those without Euler products must have zeros within the region of absolute convergence. For instance, we prove that if f is a classical holomorphic modular form whose L-function does not vanish for Re(s) > (k+1)/2, then f is a Hecke eigenform. Our proof adapts and extends work of Saias and Weingartner, who proved a similar result for degree 1 L-functions.

The universality for linear combinations of Lerch zeta functions and the Tornheim–Hurwitz type of double zeta functions

In this paper, we consider the universality for linear combinations of Lerch zeta functions. J. Kaczorowski, A. Laurincikas and J. Steuding treated universal Dirichlet series with the case that the compact sets \({\mathcal{K}_l}\) are disjoint. But we consider the both cases that the compact subset \({\mathcal{K}_l}\) is disjoint and not disjoint. Next, we will show the non-trivial zeros of the Tornheim–Hurwitz type of double zeta functions in the region of absolute convergence. Moreover we show the universality for the Tornheim–Hurwitz type of double zeta function.

A class of fixed denominator rational integrators

Existing methods such as Niekerk (1987) and Fatunla (1982) are effective but could not handle problem whose initial value is zero. In this paper, we designed and implemented a class of free denominator rational integrators for the solution of initial value problems (IVPs) in ordinary differential equations (ODES), particularly for the case of Stiff and singular problems. The class of integration obtained was found to be consistent and convergent. Our study of the stability characteristics reveal that the integrators are A-stable when m = 0, 1, 2. Their Region of Absolute Stability (RAS) decreases with increasing value of m. Experiments carried out and analyzed with the computer shows encouraging computational results and also show that the rational integrator copes well with all kinds of problem. Key words: Fixed denominator, rational integrators, Jordan curves, A-stable Region of Absolute stability, consistency and convergence.

A quadratic based integration scheme for the solution of singulo-stiff differential equations

In this term paper, we designed a quadratic based integration scheme for the solution of initial value problem (IVPs) in ordinary differential equation (ODEs). This was achieved by considering the rational interpolating operator satisfying A class of rational integrator formula given by in Aashikpelokhai (1991) with K=14 was also implemented and the results compared. The vector q= (q1, q2………qk) are obtained from the simultaneous linear equation (SLE) Sq=b where P0 = yn, The results as analyzed with the computer show that the integrator copes favourably well with singular problems, stiff problem and singulo-stiff problems. Keywords: Rational Integrators, region of absolute stability, Integration Scheme, Consistency, convergence, stiff and singular problems.

Characteristics of power growth of a multidimensional series of exponents

The behavior of sums of multidimensional series of exponents near the boundary of the region of absolute convergence is studied.

Turán inequalities and zeros of Dirichlet series associated with certain cusp forms

The "Turan inequalities" are a countably infinite set of conditions about the power series coefficients of certain entire functions which are necessary in order for the function to have only real zeros. We give a one-parameter family of generalized Dirichlet series, each with functional equation, for which the Turan inequalities hold for the associated ξ \xi -function (normalized so that the critical line is the real axis). For a discrete set of values of the parameter the Dirichlet series has an Euler product and is the L-series associated to a modular form. For these we expect the analogue of the Riemann Hypothesis to hold. For the rest of the values of the parameter we do not expect an analogue of the Riemann Hypothesis. We show for one particular value of the parameter that the Dirichlet series in fact has zeros within the region of absolute convergence.

Stochastic Born Series for Sound Propagation in Random Media. I. The Expectation of the Field

The propagation of acoustic waves in stochastic media is a problem of fundamental importance in the study of the stability of such media. This paper represents a first attempt to construct a full Born series for the expectation of the potential function. Keller's second-order results are obtained as a special case. An example, with no particular application in mind other than to establish a bound on the region of absolute convergence, is used to illustrate the theory. Higher-order moments of the solution, as well as other examples, are best left as the subjects of other papers.