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In this paper, we study a family of single variable integral representations for some products of $\zeta(2n+1)$, where $\zeta(z)$ is Riemann zeta function and $n$ is positive integer. Such representation involves the integral…

Two identities extracted from the literature are coupled to obtain an integral equation for Riemann's $\xi(s)$ function, and thus $\zeta(s)$ indirectly. The equation has a number of simple properties from which useful derivations flow, the…

Let $\zeta(s)$ and $Z(t)$ be the Riemann zeta function and Hardy's function respectively. We show asymptotic formulas for $\int_0^T Z(t)\zeta(1/2+it)dt$ and $\int_0^T Z^2(t) \zeta(1/2+it)dt$. Furthermore we derive an upper bound for…

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,…

In this article, we derive a Euler prime product formula for the magnitude of the Riemann zeta function $\zeta(s)$ valid for $\Re(s)>1$, as well as similar formulas for $\zeta(s)$ valid for an even and odd $k$th positive integer argument.…

We revisit a representation for the Riemann zeta function $\zeta(s)$ expressed in terms of normalised incomplete gamma functions given by the author and S. Cang in Methods Appl. Anal. {\bf 4} (1997) 449--470. Use of the uniform asymptotics…

The author derives new family of series representations for the values of the Riemann Zeta function $\zeta(s)$ at positive odd integers. For $n\in\mathbb{N}$, each of these series representing $\zeta(2n+1)$ converges remarkably rapidly with…

In this paper, we introduce a class of Dirichlet series defined in terms of the Riemann zeta-function, motivated by the study of their special values, and establish integral representations for these series. We also define an extension of…

We intimate deeper connections between the Riemann zeta and gamma functions than often reported and further derive a new formula for expressing the value of $\zeta(2n+1)$ in terms of zeta at other fractional points. This paper also…

We give closed-form expressions for the Dirichlet beta function at even positive integers and for the Dirichlet lambda function at odd positive integers, based on the function J(s) defined via convergent integral. We also show fundamental…

We supplement a very recent paper of R. Crandall concerned with the multiprecision computation of several important special functions and numbers. We show an alternative series representation for the Riemann and Hurwitz zeta functions…

By using the method of iterated integral representations of series, we establish some explicit relationships between multiple zeta values and Integrals of logarithmic functions. As applications of these relations, we show that multiple zeta…

In this work we introduce a new polynomial representation of the Bernoulli numbers in terms of polynomial sums allowing on the one hand a more detailed understanding of their mathematical structure and on the other hand provides a…

Assuming the Generalized Riemann Hypothesis, we establish explicit bounds in the $q$-aspect for the logarithmic derivative $\left(L'/L\right)\left(\sigma,\chi\right)$ of Dirichlet $L$-functions, where $\chi$ is a primitive character modulo…

From eigensolutions of the harmonic oscillator or Kepler-Coulomb Hamiltonian we extend the functional equation for the Riemann zeta function and develop integral representations for the Riemann xi function that is the completed classical…

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…

An analog of the Riemann hypothesis is proved in this paper. Some new integral equations for the functions $\pi(x)$ and $R(x)$ follows. A new effect that is shown is that these function - with essentially different behavior - are the…

We compute the number of irreducible linear representations of self-similar branch groups, by expressing these numbers as the co\"efficients a_n of a Dirichlet series sum a_n n^{-s}. We show that this Dirichlet series has a positive…

The main objective of this paper is to introduce a new extension of Hurwitz-Lerch Zeta function in terms of extended beta function. We then investigate its important properties such as integral representations, differential formulas, Mellin…

We review generalized zeta functions built over the Riemann zeros (in short: "superzeta" functions). They are symmetric functions of the zeros that display a wealth of explicit properties, fully matching the much more elementary Hurwitz…