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

An Elementary Proof of ∑ and a Recurrence Formula for 2k ζ

In this note, a series expansion technique introduced recently by Dancs and He for generating Euler-type formulae for odd zeta values () 2 1 k ζ + , () s ζ being the Riemann zeta function and k a positive integer, is modified in a manner to furnish the even zeta values () 2k ζ. As a result, we find an elementary proof of 2 2 1 1/ / 6 n n π ∞ = = ∑ , as well as a recurrence formula for () 2k ζ from which it follows that the ratio () 2 2 k k ζ π is a rational number, without making use of Euler's formula and Bernoulli numbers.

An Euler-type formula for β (2 n ) and closed-form expressions for a class of zeta series

Integral Transforms and Special Functions, 2012

In a recent work, Dancs and He found an Euler-type formula for ζ(2 n + 1), n being a positive integer, which contains a series they could not reduce to a finite closed-form. This open problem reveals a greater complexity in comparison to ζ(2n), which is a rational multiple of π 2n. For the Dirichlet beta function, the things are 'inverse': β(2n + 1) is a rational multiple of π 2n+1 and no closed-form expression is known for β(2n). Here in this work, I modify the Dancs-He approach in order to derive an Euler-type formula for β(2n), including β(2) = G, the Catalan's constant. I also convert the resulting series into zeta series, which yields new exact closed-form expressions for a class of zeta series involving β(2n) and a finite number of odd zeta values. A closed-form expression for a certain zeta series is also conjectured.

A series representation for Riemann's zeta function and some interesting identities that follow

Journal of Classical Analysis, 2021

Using Cauchy's Integral Theorem as a basis, what may be a new series representation for Dirichlet's function η(s), and hence Riemann's function ζ(s), is obtained in terms of the Exponential Integral function E s (iκ) of complex argument. From this basis, infinite sums are evaluated, unusual integrals are reduced to known functions and interesting identities are unearthed. The incomplete functions ζ ± (s) and η ± (s) are defined and shown to be intimately related to some of these interesting integrals. An identity relating Euler, Bernouli and Harmonic numbers is developed. It is demonstrated that a known simple integral with complex endpoints can be utilized to evaluate a large number of different integrals, by choosing varying paths between the endpoints.

On Ramanujan's formula for ζ(1/2) and ζ(2m + 1)

Journal of Mathematical Analysis and Applications, 2021

Page 332 of Ramanujan's Lost Notebook contains a compelling identity for ζ(1/2), which has been studied by many mathematicians over the years. On the same page, Ramanujan also recorded the series, 1 r exp(1 s x) − 1 + 2 r exp(2 s x) − 1 + 3 r exp(3 s x) − 1 + • • • , where s is a positive integer and r −s is any even integer. Unfortunately, Ramanujan doesn't give any formula for it. This series was rediscovered by Kanemitsu, Tanigawa, and Yoshimoto, although they studied it only when r − s is a negative even integer. Recently, Dixit and the second author generalized the work of Kanemitsu et al. and obtained a transformation formula for the aforementioned series with r − s is any even integer. While extending the work of Kanemitsu et al., Dixit and the second author obtained a beautiful generalization of Ramanujan's formula for odd zeta values. In the current paper, we investigate transformation formulas for an infinite series, and interestingly, we derive Ramanujan's formula for ζ(1/2), Wigert's formula for ζ(1/k) as well as Ramanujan's formula for ζ(2m + 1). Furthermore, we obtain a new identity for ζ(−1/2) in the spirit of Ramanujan.

Certain classes of series associated with the Zeta and related functions

Applied Mathematics and Computation, 2003

A fascinatingly large number of seemingly independent solutions of the so-called Basler problem of evaluating the Riemann Zeta function fðsÞ when s ¼ 2, which was of vital importance to Leonhard Euler (1707-1783) and the Bernoulli brothers (Jakob Bernoulli (1654-1705) and Johann Bernoulli (1667-1748)), have appeared in the mathematical literature ever since Euler first solved this problem in the year 1736. The main object of this two-part series of lectures is to present some recent developments on the evaluations and representations of fðsÞ when s 2 N n f1g, N being the set of natural numbers. We emphasize upon several interesting classes of rapidly convergent series representations for fð2n þ 1Þ ðn 2 NÞ which have been developed in recent years. In two of many computationally useful special cases considered here, it is observed that fð3Þ can be represented by means of series which converge much more rapidly than that in EulerÕs celebrated formula as well as the series used recently by Roger Ap e ery (1916-1994) in his proof of the irrationality of fð3Þ. Symbolic and numerical computations using Mathematica (Version 4.0) for Linux show, among other things, that only 50 terms of one of these series are capable of producing an accuracy of seven decimal places.

The Riemann zeta function and classes of infinite series

Applicable Analysis and Discrete Mathematics, 2017

∞ n=2 n even ζ(n) n s n and we apply our results to obtain new representations for some mathematical constants such as the Euler (or Euler-Mascheroni) constant, the Catalan constant, log 2, ζ(3) and π.