Related papers: Integral formulas for a Dirichlet series
In this paper we study the convergence of multiple Dirichlet L-series. In particular we give an integral representation of the series in the region of convergence by using Abel's summation formula. A certain generalized result is also…
Using the Dirichlet integrals, which are employed in the theory of Fourier series, this paper develops a useful method for the summation of series and the evaluation of integrals.
In this paper, We use the Fourier series expansion of real variables function, We give a formula to calculate the Dirichlet character sum, and four special examples are given.
We derive integral representations for six families of multiple Ap\'ery-like series using repeated integration by parts and Fourier expansions. The resulting formulas are expressed in terms of polylogarithms, Legendre chi functions, and…
We generalize techniques of Addison to a vastly larger context. We obtain integral representations in terms of the first periodic Bernoulli polynomial for a number of important special functions including the Lerch zeta, polylogarithm,…
This paper provides a Liouville principle for integration in terms of dilogarithm and partial result for polylogarithm.
The problem of representation of elements of weighted space of infinitely differentiable functions on real line by exponential series is considered.
This paper presents a number of identities for Dirichlet series and series with Stirling numbers of the first kind. As coefficients for the Dirichlet series we use Cauchy numbers of the first and second kinds, hyperharmonic numbers,…
By using Cauchy integral formula in the theory of complex functions, the authors establish some integral representations for the principal branches of several complex functions involving the logarithmic function, find some properties, such…
In this paper we present a special formula for transforming integrals to series. The resulting series involves binomial transforms with the Taylor coefficients of the integrand. Five applications are provided for evaluating challenging…
We consider a sum of the derivatives of Dirichlet $L$-functions over the zeros of Dirichlet $L$-functions. We give an asymptotic formula for the sum.
In the present article, the author uses Fourier theory of tempered distributions (generalized functions) in deriving a formula for Dirichlet-like integrals. The applied method is remarkably efficient and allows a solution in a few…
We obtain a new decomposition of the Riemann-Liouville operators of fractional integration as a series involving derivatives (of integer order). The new formulas are valid for functions of class $C^n$, $n \in \mathbb{N}$, and allow us to…
The manuscript reviews Dirichlet Series of important multiplicative arithmetic functions. The aim is to represent these as products and ratios of Riemann zeta-functions, or, if that concise format is not found, to provide the leading…
In this rather computational paper, we determine certain representation numbers of ideals in real quadratic number fields explicitly in order to obtain a representation of the associated Dirichlet series in terms of Dirichlet L-functions…
As a continuation of the authors and Wakatsuki's previous paper [5], we study relations among Dirichlet series whose coefficients are class numbers of binary cubic forms. We show that for any integral models of the space of binary cubic…
The Riemann-Liouville fractional integrals and derivatives are generalized for cases when fractional exponent $d$ are functions of space and times coordinates (i.e. $d=d({\bf r}(t),t)$).
We define the derivative of an integer to be the map sending every prime to 1 and satisfying the Leibniz rule. The aim of this article is to calculate the Dirichlet product of this map with a function arithmetic multiplicative.
This paper provides a Liouville principle for integration in terms of exponential integrals and incomplete gamma functions.
We derive a general formula for the product of two Dirichlet series that satisfy Hecke's functional equation. Several examples are provided to demonstrate the applicability of the formula. In addition, we discuss prior work on similar…