Related papers: On Hilbert-Polya conjecture: Hermitian operator na…

We define an analogue of the classical Mittag-Leffler function which is applied to two variables, and establish its basic properties. Using a corresponding single-variable function with fractional powers, we define an associated fractional…

Riemann's hypothesis, formulated in 1859, concerns the location of the zeros of Riemann's Zeta function. The history of the Riemann hypothesis is well known. In 1859, the German mathematician B. Riemann presented a paper to the Berlin…

From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical…

We show that all Hankel operators $H$ realized as integral operators with kernels $h(t+s)$ in $L^2 ({\Bbb R}_{+}) $ can be quasi-diagonalized as $H= {\sf L}^* \Sigma {\sf L} $. Here ${\sf L}$ is the Laplace transform, $\Sigma$ is the…

This paper is divided into two independent parts. The first part presents new integral and series representations of the Riemaan zeta function. An equivalent formulation of the Riemann hypothesis is given and few results on this formulation…

We revisit and connect several notions of algebraic multiplicities of zeros of analytic operator-valued functions and discuss the concept of the index of meromorphic operator-valued functions in complex, separable Hilbert spaces.…

This paper gives a representation-theoretic interpretation of the Lerch zeta function and related Lerch $L$-functions twisted by Dirichlet characters. These functions are associated to a four-dimensional solvable real Lie group $H^{J}$,…

In this article, we explore a natural extension of the quadratic parametrization introduced in our previous work. By replacing the integer $n$ by $n^s$ ($ s\in\mathbb{R}, s>1$) and allowing the parameters to be real, we obtain for each…

We first construct a dynamical systems model which in its steady-state serves as an analytic continuation of the completed Riemann zeta function over the entire critical strip. The resulting mathematical construct involves a linear…

Finite Cartesian products of operators play a central role in monotone operator theory and its applications. Extending such products to arbitrary families of operators acting on different Hilbert spaces is an open problem, which we address…

We numerically study the statistical properties of differences of zeros of Riemann zeta function and L-functions predicted by the theory of the e\~ne product. In particular, this provides a simple algorithm that computes any non-real…

A generalized modular relation of the form $F(z, w, \alpha)=F(z, iw,\beta)$, where $\alpha\beta=1$ and $i=\sqrt{-1}$, is obtained in the course of evaluating an integral involving the Riemann $\Xi$-function. It is a two-variable…

We introduce a Bernoulli operator,let $\mathbf{B}$ denote the operator symbol,for n=0,1,2,3,... let ${\mathbf{B}^n}: = {B_n}$ (where ${B_n}$ are Bernoulli numbers,${B_0} = 1,B{}_1 = 1/2,{B_2} = 1/6,{B_3} = 0$...).We obtain some formulas for…

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…

Several arguments against the truth of the Riemann hypothesis are extensively discussed. These include the Lehmer phenomenon, the Davenport-Heilbronn zeta-function, large and mean values of $|\zeta(1/2+it)|$ on the critical line, and zeros…

In this series of seven papers, predominantly by means of elementary analysis, we establish a number of identities related to the Riemann zeta function. Whilst this paper is mainly expository, some of the formulae reported in it are…

We rewrite the zero-counting formula within the critical strip of the Riemann zeta function as a cumulative density distribution; this subsequently allows us to formally derive an integral expression for the Li coefficients associated with…

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…

We develop a duality theory for unbounded Hermitian operators with dense domain in Hilbert space. As is known, the obstruction for a Hermitian operator to be selfadjoint or to have selfadjoint extensions is measured by a pair of deficiency…

Let $A$ be a densely defined symmetric operator with equal deficiency indices in a Hilbert space. We introduce the notion of a Weyl function $M(z)$ of $A$ corresponding to an ordinary boundary triplet of the operator $A^*$ and then…