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Michael Th. Rassias

Publications and source records attributed to Michael Th. Rassias.

31 records · Page 2Linked to original sources
arXiv2015

A Hilbert-Type Integral Inequality in the Whole Plane Related to the Hypergeometric Function and the Beta Function

A new Hilbert-type integral inequality in the whole plane with the non-homogeneous kernel and parameters is given. The constant factor related to the hypergeometric function and the beta function is proved to be the best possible. As applications, equivalent forms, the reverses, some particular examples, two kinds of Hardy-type inequalities, and operator expressions are considered.

math.CA↗
arXiv2015

A Half-Discrete Hardy-Hilbert-Type Inequality with a Best Possible Constant Factor Related to the Hurwitz Zeta Function

Using methods of weight functions, techniques of real analysis as well as the Hermite-Hadamard inequality, a half-discrete Hardy-Hilbert-type inequality with multi-parameters and a best possible constant factor related to the Hurwitz zeta function and the Riemann zeta function is obtained. Equivalent forms, normed operator expressions, their reverses and some particular cases are also considered.

math.CA↗
arXiv2015

A More Accurate Half-Discrete Hardy-Hilbert-Type Inequality with the Best Possible Constant Factor Related to the Extended Riemann-Zeta Function

By the method of weight coefficients, techniques of real analysis and Hermite-Hadamard's inequality, a half-discrete Hardy-Hilbert-type inequality related to the kernel of the hyperbolic cosecant function with the best possible constant factor expressed in terms of the extended Riemann-zeta function is proved. The more accurate equivalent forms, the operator expressions with the norm, the reverses and some particular cases are also considered.

math.CA↗
arXiv2015

Parameterized Yang-Hilbert-Type Integral Inequalities and Their Operator Expressions

Applying methods of Real Analysis and Functional Analysis, we build two weight functions with parameters and provide two kinds of parameterized Yang-Hilbert-type integral inequalities with the best constant factors. Equivalent forms, the reverses, and the operator expressions are also given. In particular, the Hardy-type inequalities and Hardy-type operators are studied. Additionally, a number of examples with two kinds of particular kernels are considered.

math.FA↗
arXiv2015

Trigonometric representations of generalized Dedekind and Hardy sums via the discrete Fourier transform

We introduce some new higher dimensional generalizations of the Dedekind sums associated with the Bernoulli functions and of those Hardy sums which are defined by the sawtooth function. We generalize a variant of Parseval's formula for the discrete Fourier transform to derive finite trigonometric representations for these sums in a simple unified manner. We also consider a related sum involving the Hurwitz zeta function.

math.NT↗
arXiv2015

The order of magnitude for moments for certain cotangent sums

We settle a question on the rate of growth of the moments of cotangent sums considered by the authors in their previous papers [8], [9]. We even obtain the true order of magnitude of these moments. We include as well the moments of order $2k$ between fixed multiples of $(2k)!/π^{2k}$.

math.NT↗
arXiv2014

The rate of growth of moments of certain cotangent sums

We consider cotangent sums associated to the zeros of the Estermann zeta function considered by the authors in their previous paper [5]. We settle a question on the rate of growth of the moments of these cotangent sums left open in [5], and obtain a simpler proof of the equidistribution of these sums.

math.NT↗
arXiv2014

Generalizations of a cotangent sum associated to the Estermann zeta function

Cotangent sums are associated to the zeros of the Estermann zeta function. They have also proven to be of importance in the Nyman-Beurling criterion for the Riemann Hypothesis. The main result of the paper is the proof of the existence of a unique positive measure μ on $\mathbb{R}$, with respect to which certain normalized cotangent sums are equidistributed. Improvements as well as further generalizations of asymptotic formulas regarding the relevant cotangent sums are obtained. We also prove an asymptotic formula for a more general cotangent sum as well as asymptotic results for the moments of the cotangent sums under consideration. We also give an estimate for the rate of growth of the moments of order 2k, as a function of k.

math.NT↗
arXiv2013

A Multidimensional Hilbert-Type Integral Inequality Related to the Riemann Zeta Function

In this paper,using methods of weight functions and techniques of real analysis, we provide a multidimensional Hilbert-type integral inequality with a homogeneous kernel of degree 0 as well as a best possible constant factor related to the Riemann zeta function. Some equivalent representations and certain reverses are obtained. Furthermore, we also consider operator expressions with the norm and some particular results.

math.NT↗
arXiv2013

On the representation of the number of integral points of an elliptic curve modulo a prime number

In this paper we shall investigate the problem of the representation of the number of integral points of an elliptic curve modulo a prime number p. We present a way of expressing an exponential sum which involves polynomials of third degree, in explicit non-exponential terms. In the process, we present explicit formulas for the calculation of some series involving the Riemann Zeta function.

math.NT↗