Related papers: Vector-valued general Dirichlet series

This paper contains a development of the Theory of Lebesgue and Bochner spaces of summable functions. It represents a synthesis of the results due to H. Lebesgue, S. Banach, S. Bochner, G. Fubini, S. Saks, F. Riesz, N. Dunford, P. Halmos,…

Given a frequency sequence $\omega=(\omega_n)$ and a finite subset $J \subset \mathbb{N}$, we study the space $\mathcal{H}_{\infty}^{J}(\omega)$ of all Dirichlet polynomials $D(s) := \sum_{n \in J} a_n e^{-\omega_n s}, \, s \in \mathbb{C}$.…

We introduce a new functional space U designed to contain all classical arithmetic functions (Mobius, von Mangoldt, Euler phi, divisor functions, Dirichlet characters, etc.). The norm of U combines a Hilbert-type component, based on square…

We study the triple convolution sum of the generalised divisor functions $$\sum_{n\leq x} d_k(n+h)d_l(n)d_m(n-h),$$ where $h \le x^{1-\epsilon}$ for any $\epsilon>0$ and $d_k(n)$ denotes the generalised divisor function which counts the…

Let $p_1,p_2,\dots,p_n, a_1,a_2,\dots,a_n \in \N$, $x_1,x_2,\dots,x_n \in \R$, and denote the $k$th periodized Bernoulli polynomial by $\B_k(x)$. We study expressions of the form \[ \sum_{h \bmod{a_k}} \ \prod_{\substack{i=1\\ i\not=k}}^{n}…

Direct links between generalized harmonic numbers, linear Euler sums and Tornheim double series are established in a more perspicuous manner than is found in existing literature. We show that every linear Euler sum can be decomposed into a…

The paper proves a generalization of Wintner's theorem on the asymptotics of summation functions to the case of summation functions with nonlinear asymptotics. The class of arithmetic functions that have a logarithmic asymptotic mean is…

Sequence transformations are valuable numerical tools that have been used with considerable success for the acceleration of convergence and the summation of diverging series. However, our understanding of their theoretical properties is far…

Stochastic linear combinations of some random vectors are studied where the distribution of the random vectors and the joint distribution of their coefficients are Dirichlet. A method is provided for calculating the distribution of these…

Motivated by the study of the distribution of zeros of generalized Bessel-type functions, the principal goal of this paper is to identify new research directions in the theory of multiplier sequences. The investigations focus on multiplier…

After different variables and functions changes, the generalized dispersal problem, recalled in (1) below and considered in part I, see Labbas, Maingot and Thorel [14], leads us to consider, to study and to invert the sum of linear…

We characterize the uniform limits of Dirichlet polynomials on a right half plane. In the Dirichlet setting, we find approximation results, with respect to the Euclidean distance and {to} the chordal one as well, analogous to classical…

The space $\widehat{\ell }_k$ of absolutely almost convergent series was introduced and studied by Das et al [4]; which plays an important role in summability theory, approximation theory, Fourier analysis, etc. In the present paper we…

We survey some recent developments in the analytic theory of multiple Dirichlet series with arithmetical coefficients on the numerators.

Our main result is to answer a question of Michel Balazard by giving a Dirichlet series with only one zero in its half-plane of convergence. At the end of the paper we also give several sufficient conditions for the Generalized Riemann…

In this paper, we study the notion of mid summability in a general setting using the duality theory of sequence spaces. We define the vector valued sequence space $\lambda^{mid}(X)$ corresponding to a Banach space $X$ and sequence space…

We study three special Dirichlet series, two of them alternating, related to the Riemann zeta function. These series are shown to have extensions to the entire complex plane and we find their values at the negative integers (or residues at…

Hartman proved in 1939 that the width of the largest possible strip in the complex plane, on which a Dirichlet series $\sum_n a_n n^{-s}$ is uniformly a.s.-sign convergent (i.e., $\sum_n \varepsilon_n a_n n^{-s}$ converges uniformly for…

A few years ago, the concept of a D-analogue was introduced as a Dirichlet series analogue for the already known and well researched hypergeometric q-series. The D-analogue of the q-Dixon sum is given here, in the context of seeing a direct…

We consider a Dirichlet series $\sum_{n=1}^{\infty}a_n^{-s}$, where $a_n$ satisfies a linear recurrence of arbitrary degree with integer coefficients. Under suitable hypotheses, we prove that it has a meromorphic continuation to the complex…