Related papers: Harmonic Analysis of some arithmetical functions

In this paper we analyze some classical operators in harmonic analysis associated to the multidimensional discrete Laplacian \[ \Delta_N f(\mathbf{n})=\sum_{i=1}^{N}(f(\mathbf{n}+\mathbf{e}_i)-2f(\mathbf{n})+f(\mathbf{n}-\mathbf{e}_i)),…

Using Euler transformation of series we relate values of Hurwitz zeta function at integer and rational values of arguments to certain rapidly converging series where some generalized harmonic numbers appear. The form of these generalized…

Toric ideals to hierarchical models are invariant under the action of a product of symmetric groups. Taking the number of factors, say m, into account, we introduce and study invariant filtrations and their equivariant Hilbert series. We…

We define a special function related to the digamma function and use it to evaluate in closed form various series involving binomial coefficients and harmonic numbers.

In this technical report, certain interesting classification of arithmetical functions is proposed. The notion of additively decomposable and multiplicatively decomposable arithmetical functions is proposed. The concepts of arithmetical…

We show that integrals involving log-tangent function, with respect to certain square-integrable functions on $(0, \pi/2)$, can be evaluated by some series involving the harmonic number. Then we use this result to establish many closed…

We investigate the properties of the moments of the cot function using the central factorial numbers. Using a new integral representation of the central factorial numbers, we find a new way to express these moments in terms of recursive…

We attach a certain $n \times n$ matrix $A_n$ to the Dirichlet series $L(s)=\sum_{k=1}^{\infty}a_k k^{-s}$. We study the determinant, characteristic polynomial, eigenvalues, and eigenvectors of these matrices. The determinant of $A_n$ can…

This paper is a continuation of our previous work on double Dirichlet series associated with arithmetic functions such as the von Mangoldt function, the M\"obius function, and so on. We consider the analytic behaviour around the…

We~describe the Dirichlet space of $M$-harmonic functions, i.e.~functions annihilated by the invariant Laplacian on~the unit ball of the complex $n$-space, as~the limit of the analytic continuation (in~the spirit of Rossi and Vergne) of the…

We present a large number of analytic evaluations of Euler sums, namely sums such as \begin{align} M(m,n_0,n_1,n_2, \ldots, n_t) &= \sum_{k=1}^\infty \frac{H(k)^m}{k^{n_0} (k+1)^{n_1} (k+2)^{n_2} \cdots (k+t)^{n_t}}, \nonumber \end{align}…

We characterize the set of positive harmonic functions with Dirichlet boundary conditions in unbounded domains which are union of several different chambers. We analyze the asymptotic behavior of the solutions in connection with the changes…

This note gives a few rapidly convergent series representations of the sums of divisors functions. These series have various applications such as exact evaluations of some power series, computing estimates and proving the existence results…

The purpose of this work is to present a class of harmonic univalent functions defined by the Dziok-Srivastava operator. Some geometric properties like coefficients conditions, distortion theorem, convolution (Hadamard product), convex…

This paper is a contribution to the study of the relations between special functions, Lie algebras and rigged Hilbert spaces. The discrete indices and continuous variables of special functions are in correspondence with the representations…

We characterize the space of multipliers from the Hardy space of Dirichlet series $\mathcal H_p$ into $\mathcal H_q$ for every $1 \leq p,q \leq \infty$. For a fixed Dirichlet series, we also investigate some structural properties of its…

The Dirichlet--Hardy space $\Ht$ consists of those Dirichlet series $\sum_n a_n n^{-s}$ for which $\sum_n |a_n|^2<\infty$. It is shown that the Blaschke condition in the half-plane $\operatorname{Re} s>1/2$ is a necessary and sufficient…

Summation by parts is used to find the sum of a finite series of generalized harmonic numbers involving a specific polynomial or rational function. The Euler-Maclaurin formula for sums of powers is used to find the sums of some finite…

Harmonic functions of two variables are exactly those that admit a conjugate, namely a function whose gradient has the same length and is everywhere orthogonal to the gradient of the original function. We show that there are also partial…

In this paper we begin the study of some important Banach spaces of slice hyperholomorphic functions, namely the Bloch, Besov and weighted Bergman spaces, and we also consider the Dirichlet space, which is a Hilbert space. The importance of…