On some expansions for the Euler Gamma function and the Riemann Zeta function

Some new inequalities for Generalized Mathieu type series and Riemann zeta functions

Our aim in this paper is to show some new inequalities for Mathieu's type series and Riemann zeta function. In particular, some Turán type inequalities, some monotonicity and log-convexity results for these special functions are given. New Laplace type integral representations for Mathie type series and Riemann zeta function are also presented.

On some properties of Riemann zeta function on critical line

The aim of this paper is to show further results following those published in [5], and to relate the Riemann zeta function to the relativistic cosmology.

Special values and integral representations for the Hurwitz-type Euler zeta functions

The Hurwitz-type Euler zeta function is defined as a deformation of the Hurwitz zeta function: \begin{equation*} ζ_E(s,x)=\sum_{n=0}^\infty\frac{(-1)^n}{(n+x)^s}. \end{equation*} In this paper, by using the method of Fourier expansions, we shall evaluate several integrals with integrands involving Hurwitz-type Euler zeta functions $ζ_E(s,x)$. Furthermore, the relations between the values of a class of the Hurwitz-type (or Lerch-type) Euler zeta functions at rational arguments have also been given.

A signed analog of Euler's reduction formula for the double zeta function

The double zeta function is a function of two arguments defined by a double Dirichlet series, and was first studied by Euler in response to a letter from Goldbach in 1742. By calculating many examples, Euler inferred a closed form evaluation of the double zeta function in terms of values of the Riemann zeta function, in the case when the two arguments are positive integers with opposite parity. Here, we consider a signed analog of Euler's evaluation: namely a reduction formula for the signed double zeta function that reduces to Euler's evaluation when the signs are specialized to 1. This formu…

On Gram's law in the theory of the Riemann zeta function

Some Titchmarsh results following Gram's law are improved in this paper.

On Müntz-type formulas related to the Riemann zeta function

The Mellin transform and several Dirichlet series related with the Riemann zeta function are used to deduce some identities similar to the classical Müntz formula [4]. These formulas are derived in the critical strip and in the half-plane $Re(s)<0$. As particular cases, integral representations for products of the gamma and zeta functions are exhibited.

On some mellin transforms for the Riemann zeta function in the critical strip

We offer two new Mellin transform evaluations for the Riemann zeta function in the region $0<\Re(s)<1.$ Some discussion is offered in the way of evaluating some further Fourier integrals involving the Riemann xi function.

New rapidly converging series representations for values of the Riemann zeta function and the Dirichlet beta function

In this paper we derive rapidly converging series for Catalan's constant and for Apéry's constant. The method may be easily generalised to produce new series representations for other values of the Riemann zeta function and the Dirichlet beta function.

Some Additions to a Family of Sums and Integrals related to Hurwitz' Zeta Function(s), Euler polynomials and Euler Numbers

Integrals involving the kernel function $sech (πx)$ over a semi-infinite range are of general interest in the study of Riemann's function $ζ(s)$ and Hurwitz' function $ζ(s,a)$. Such integrals that include the $arctan$ and $log$ functions in the integrand are evaluated here in terms of $ζ(s,a)$, thereby adding some new members to a known family of related integrals. A claimed connection between $ζ(s)$ of odd integer argument and such integrals is verified.

Limit Representations of Riemann's Zeta Function

In this paper it is shown that Riemann's zeta function $ζ(s)$ admits two limit representations when $\Re{(s)}>1.$ Each of these limit representations is deduced by using simple arguments based upon the classical Tannery's (limiting) theorem for series.

A new integral representation for the Riemann Zeta function

A new integral representation for the Riemann zeta function is derived. This representation covers the important region of the complex plane where the real part of the argument of the function lies between 0 and 1. Using this representation, we obtain new functional identities for the Riemann zeta function.

Elegant expressions and generic formulas for the Riemann zeta function for integer arguments

A new definition for the Riemann zeta function for all positive integer number s > 1 is presented. We discover a most elegant expression and easy method for calculating the Riemann zeta function for small even integer values. Through this new reformulation we provide a one-line proof of the value of zeta(2) and demonstrate that zeta(2s) may be calculated by hand using only the cotangent function when the magnitude of the integer s is small.

Fundamental relations between the Dirichlet beta function, euler numbers, and Riemann zeta function for positive integers

A new definition for the Dirichlet beta function for positive integer arguments is discovered and presented for the first time. This redefinition of the Dirichlet beta function, based on the polygamma function for some special values, provides a general method for obtaining all special constants associated with Dirichlet beta function. We also show various new and fundamental relations between the polygamma function, Riemann zeta, the even-indexed euler numbers, the Dirichlet beta functions in a way never seen or imagined before.

Notes on a paper of Tyagi and Holm: A new integral representation for the Riemann Zeta function

It is shown that a new series representation of Riemann s Zeta function obtained by Tyagi and Holm leads to an interesting new recursion for Bernoulli numbers of even index as well as new representations of, and infinite series involving, Zeta functions of special (integer) argument.

Bessel函数级数作为Riemann zeta函数级数的表示

Series over Bessel functions as series in terms of Riemann's zeta function

本文利用Hurwitz公式和三角级数求和公式，将Bessel函数级数转化为Riemann zeta函数级数，并求得有限和。

黎曼ζ函数 $ζ(s)$ 的一种新表示

A New Representation of the Riemann Zeta Function $ζ(s)$

本文推广了黎曼ζ函数与伯努利数的已知关系，提出了一种以伯努利数嵌套级数表示ζ函数的新公式。