Related papers: Addison-type series representation for the Stieltj…

We give a continued-fraction characterization of Stieltjes moment sequences for which there exists a representing measure with support in $[\xi, \infty)$. The proof is elementary.

In this paper, we study a Dirichlet series generated by powers of harmonic numbers. As an application of these functions, we derive certain series involving harmonic numbers. We also study the analytic properties of these Dirichlet series…

In this article, we develop a square-free zeta series associated with the M\"obius function into a power series, and prove a Stieltjes like formula for these expansion coefficients. We also investigate another analytical continuation of…

Using a different approach, we derive integral representations for the Riemann zeta function and its generalizations (the Hurwitz zeta, $\zeta(-k,b)$, the polylogarithm, $\mathrm{Li}_{-k}(e^m)$, and the Lerch transcendent,…

Using three basic facts concerning Hurwitz zeta function,we give new natural proofs of the known results on Bernoulli polynomials,gamma function and also obtain Gauss' expression for Psi function at a rational point,all in a unified…

We represent the Riemann zeta function in the half-plane $\Re s >1$ via series whose terms admit geometrically decreasing bounds. Due to an underlying recurrence relation, which is used to compute coefficients entering into the terms, the…

The classical Mertens' formula states that $ \prod_{p\le N}\big(1-\frac1p)^{-1}\sim e^\gamma\log N, $ where the product is over all primes $p$ less than or equal to $N$, and $\gamma$ is the Euler-Mascheroni constant. By the Euler product…

We compute the Lyapunov exponent, generalized Lyapunov exponents and the diffusion constant for a Lorentz gas on a square lattice, thus having infinite horizon. Approximate zeta functions, written in terms of probabilities rather than…

Relying on the Hurwitz formula, we find sums of the series over sine and cosine functions through the Hurwitz zeta function. Using another summation formula for these trigonometric series, we find finite sums of some series over the Riemann…

The renormalization of MZV was until now carried out by algebraic means. We show that renormalization in general, of the multiple zeta functions in particular, is more than mere convention. We show that simple calculus methods allow us to…

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…

We extend the theory of local constants to l-adic families of representations of GL_n(F) where F is a p-adic field with l not equal to p. We construct zeta integrals and gamma factors for representations coming from the conjectural "local…

The Riemann zeta identity at even integers of Lettington, along with his other Bernoulli and zeta relations, are generalized. Other corresponding recurrences and determinant relations are illustrated. Another consequence is the application…

Using Euler transformation of series we relate values of Hurwitz zeta function at integer and rational values of arguments to certain rapidly converging series where some generalized harmonic numbers appear. The form of these generalized…

We provide new representations for the finite parts at the poles and the derivative at zero of the Barnes zeta function in any dimension in the general case. These representations are in the forms of series and limits. We also give an…

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.

We consider the random continued fraction S(t) := 1/(s_1 + t/(s_2 + t/(s_3 + >...))) where the s_n are independent random variables with the same gamma distribution. For every realisation of the sequence, S(t) defines a Stieltjes function.…

Neither the Euler-Mascheroni constant, $\gamma=0.577215...$, nor the Euler-Gompertz constant, $\delta=0.596347...$, is currently known to be irrational. However, it has been proved that at least one of them is transcendental. The two…

A generalization of a well-known relation between the Riemann zeta function $\zeta(s)$ and Bernoulli numbers $B_n$ is obtained. The formula is a new representation of the Riemann zeta function in terms of a nested series of Bernoulli…

Let $$ \zeta_E(s,q)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+q)^{s}} $$ be the alternating Hurwitz (or Hurwitz-type Euler) zeta function. In this paper, we obtain the following asymptotic expansion of $\zeta_{E}(s,q)$ $$ \zeta_E(s,q)\sim\frac12…