Michael Th. Rassias

A half-discrete Hilbert-type inequality in the whole plane related to the Riemann zeta function

Abstract By the use of Hermite–Hadamard’s inequality and weight functions, a half-discrete Hilbert-type inequality in the whole plane with the kernel of hyperbolic cotangent function and multi-parameters is given. The constant factor related to the Riemann zeta function is proved to be the best possible. The equivalent forms, two kinds of particular inequalities, the operator expressions and some equivalent reverses are considered.

Solved and Unsolved Problems

The present column is devoted to Partial Differential Equations (PDEs). The study of PDEs has proved to have a tremendously wide spectrum of applications to various domains, from the study of black holes to mathematical finance. Such equations can be used to describe and quantitatively investigate various and diverse phenomena such as heat, sound, elasticity, fluid dynamics, quantum mechanics, etc.

Maier’s Matrix Method and Irregularities in the Distribution of Prime Numbers

This paper is devoted to irregularities in the distribution of prime numbers. We describe the development of this theory and the relation to Maier’s matrix method.

On a Hilbert-Type Integral Inequality in the Whole Plane

By using methods of real analysis and weight functions, we prove a new Hilbert-type integral inequality in the whole plane with non-homogeneous kernel and a best possible constant factor. As applications, we also consider the equivalent forms, some particular cases and the operator expressions.

The Ternary Goldbach Problem with a Prime and Two Isolated Primes

In the present paper we prove that under the assumption of the GRH (Generalized Riemann Hypothesis) each sufficiently large odd integer can be expressed as the sum of a prime and two isolated primes.

Basic Steps of the Proof of Schnirelmann’s Theorem

In this chapter we provide an outline of the proof of Schnirelmann’s theorem which states that there exists a positive integer q, such that every integer greater than 1 can be represented as the sum of at most q prime numbers.

Step-by-Step Proof of Vinogradov’s Theorem

In the first section, we begin with some lemmas and theorems which will be useful in presenting a step-by-step proof of Vinogradov’s theorem, which states that there exists a natural number N, such that every odd positive integer n, with \(n\ge N\), can be represented as the sum of three prime numbers. The experienced reader may wish to skip this section.