Related papers: On Dirichlet's lambda functions
In this paper we study restricted sum formulas involving alternating Euler sums which are defined by \zeta(s_1,...,s_{d};\epsilon_1,...,\epsilon_d)=\sum_{n_1>...>n_d\ge 1}\frac{\epsilon_1^{n_1}... \epsilon_{d}^{n_d}}{n_1^{s_1}...…
A new method for continuing the usual Dirichlet series that defines the Riemann zeta function ${\zeta}(s)$ is presented. Numerical experiments demonstrating the computational efficacy of the resulting continuation are discussed.
N. Kishore, Proc. Amer. Math. Soc. 14 (1963), 523, considered the Rayleigh functions sigma_n, sums of the negative even powers of the (non-zero) zeros of the Bessel function J_nu(z) and provided a convolution type sum formula for finding…
The paper is devoted to study analogues of the van der Corput lemmas involving Mittag-Leffler functions. The generalisation is that we replace the exponential function with the Mittag-Leffler-type function, to study oscillatory integrals…
We define the generalized-Euler-constant function $\gamma(z)=\sum_{n=1}^{\infty} z^{n-1} (\frac{1}{n}-\log \frac{n+1}{n})$ when $|z|\leq 1$. Its values include both Euler's constant $\gamma=\gamma(1)$ and the "alternating Euler constant"…
Using properties of the Riemann zeta-function we propose two new large classes of evaluated series. Incidentally the first class represents integrals as generalized average on very nonuniform sequences. The second class contains inter alia…
We introduce a family of Dirichlet series associated to real quadratic number fields that generalize the ordinary Fibonacci zeta function $\sum F(n)^{-s}$, where $F(n)$ denotes the $n$th Fibonacci number. We then give three different…
Using as starting point a classical integral representation of a L-function we define a familly of two variables extended functions which are eigenfunctions of a Hermitian operator (having imaginary part of zeros as eigenvalues). This…
In the spirit of the work of Hardy-Littlewood and Lavrik, we study the Dirichlet series associated to the generalized divisor function $\sigma_{\alpha}(n):=\sum_{d|n}d^{\alpha}$. We obtain an exact identity relating the Dirichlet series…
We represent the Euler alternating series (sometimes called the "Dirichlet eta function"), and generally $(b^s-b)\zeta(s)/b^s$ for $b>1$ an integer, in the half-plane $\Re s>0$, via series dominated by geometric series, with arbitrarily…
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…
Recently, Kaneko and Tsumura introduced multiple $\widetilde{T}$-values, another kind of poly-Euler numbers and the related Arakawa-Kaneko type zeta function. It is shown that each of them satisfies similar formulas to those of multiple…
We consider some closed-form evaluations of certain infinite sums involving the Hurwitz zeta function $\zeta(s,\alpha)$ of the form \[\sum_{k=1}^\infty (\pm 1)^k k^m \zeta(s,k),\] where $m$ is a non-negative integer. For the sums with $m=0$…
In this note, we mainly show the analogue of one of Alladi's formulas over $\mathbb{Q}$ with respect to the Dirichlet convolutions involving the M\"{o}bius function $\mu(n)$, which is related to the natural densities of sets of primes by…
Recently, $\lambda$-Bernoulli and $\lambda$-Euler numbers are studied in [5, 10]. The purpose of this paper is to present a systematic study of some families of the $q$-extensions of the $\lambda$-Bernoulli and the $\lambda$-Euler numbers…
The functional relation of the Riemann z\^eta function provides us with neither the nature nor the expression of z\^eta at positive odd numbers. From the function $F(z)=\frac{z^{-2n}}{e^z-1}$, we find a functional relation involving…
We provide efficient methods to evaluate the Riemann zeta, the Lerch zeta and the Dirichlet $L$-functions. The method uses the Riemann-Siegel (RS) type formulas and a modified double exponential (MDE) quadrature method near the saddle point…
Extending the Eulerian functions, we study their relationship with zeta function of several variables. In particular, starting with Weierstrass factorization theorem (and Newton-Girard identity) for the complex Gamma function, we are…
The aim of this article is to present in a self-contained way identities arising in elementary number theory, among which the following one: $$ \sum_{d\mid n}\frac{\mu^2(d)}{\varphi(d)\,d^s}=\prod_{p\mid n}\left(1+\frac{1}{(p-1)p^s}\right).…
In this article, we derive an expression for the complex magnitude of the Dirichlet beta function $\beta(s)$ represented as a Euler prime product and compare with similar results for the Riemann zeta function. We also obtain formulas for…