Related papers: On Hurwitz zeta function and Lommel functions
By applying the inverse Mellin transform to some simple closed form identities, a number of relationships are established that connect integrals containing Riemann's and Hurwitz' zeta functions ($\zeta(s)$ and $\zeta(s,a)$) and their…
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…
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…
By employing contour integration the derivation of a generalized double finite series involving the Hurwitz-Lerch zeta function is used to derive closed form formulae in terms of special functions. We use this procedure to find special…
A modular relation of the form $F(\alpha, w)=F(\beta, iw)$, where $i=\sqrt{-1}$ and $\alpha\beta=1$, is obtained. It involves the generalized digamma function $\psi_w(a)$ which was recently studied by the authors in their work on developing…
We build on a recent paper on Fourier expansions for the Riemann zeta function. We establish Fourier expansions for certain $L$-functions, and offer series representations involving the Whittaker function $W_{\gamma,\mu}(z)$ for the…
Dating back to Euler, in classical analysis and number theory, the Hurwitz zeta function $$ \zeta(z,q)=\sum_{n=0}^{\infty}\frac{1}{(n+q)^{z}}, $$ the Riemann zeta function $\zeta(z)$, the generalized Stieltjes constants $\gamma_k(q)$, the…
The Watson-Harkins sum involving the product of the cosine and cosecant functions is extended to derive the finite sum of generalized Hurwitz-Lerch Zeta functions is derived in terms of the Hurwitz-Lerch Zeta function. A transformation…
We define a generalized class of modified zeta series transformations generating the partial sums of the Hurwitz zeta function and series expansions of the Lerch transcendent function. The new transformation coefficients we define within…
In article, we explore the secondary zeta function $Z(s)$, which is defined as a generalized zeta type of series over imaginary parts of non-trivial zeros of the Riemann zeta function $\zeta(s)$. This function has been analytically…
A quite fast proof of the functional equation of the Riemann zeta function. It is a modification of a proof usually overlooked in Titchmarsh's monograph.
In this paper we establish a new summation method by expanding $\prod_{k}(1-\frac{z}{a_{k}})^{-1}$ with two approaches: the Taylor expansion and the infinite partial fraction decomposition. Here we focus on the case when $a_{k}$ is…
We show that for a non-positive value of the first variable,Hurwitz zeta function becomes a polynomial in the second variable. We show this, using 'integration approach', instead of 'power series approach', which we had resorted to, in our…
For the Tornheim double zeta function T(s1,s2,s3) of complex variables,we obtain its functional equations,which are new.Using the calculus of r-th order derivative of zeta(s,alpha) as a function of alpha(developed in author[7])as the…
As one of the asymptotic formulas for the zeta-function, Hardy and Littlewood gave asymptotic formulas called the approximate functional equation. In 2003, R. Garunk\v{s}tis, A. Laurin\v{c}ikas, and J. Steuding (in [1]) proved the…
Several second moment and other integral evaluations for the Riemann zeta function $\zeta(s)$, Hurwitz zeta function $\zeta(s,a)$, and Lerch zeta function $\Phi(z,s,a)$ are presented. Additional corollaries that are obtained include…
For Hurwitz Zeta function,we consider its Taylor series expansion about various points as an analytic function of second variable in appropriate discs.We show that these Taylor are all polynomials in second variable for a non positive…
In this paper we construct a $\hat\mathbb{Z}$-valued measure on $\hat\mathbb{Z}$ which interpolates $p$-adic Hurwitz zeta functions for all $p$.
We prove a monotonicity property of the Hurwitz zeta function which, in turn, translates into a chain of inequalities for polygamma functions of different orders. We provide a probabilistic interpretation of our result by exploiting a…
This paper gives some results for the logarithm of the Riemann zeta-function and its iterated integrals. We obtain a certain explicit approximation formula for these functions. The formula has some applications, which are related with the…