A simple computation of  zeta(2k)\ zeta (2k)  zeta(2k) by using Bernoulli polynomials and a telescoping series (original) (raw)

A simple computation of ζ(2k) by using Bernoulli polynomials and a telescoping series

2012

We present a new proof of Euler's formulas for ζ(2k), where k = 1,2,3,..., which uses only the defining properties of the Bernoulli polynomials, obtaining the value of ζ(2k) by summing a telescoping series. Only basic techniques from Calculus are needed to carry out the computation. The method also applies to ζ(2k+1) and the harmonic numbers, yielding integral formulas for these.

A Simple Computation of ζ (2k)

The American Mathematical Monthly

We present a new simple proof of Euler's formulas for ζ(2k), where k = 1, 2, 3, . . . . The computation is done using only the defining properties of the Bernoulli polynomials and summing a telescoping series, and the same method also yields integral formulas for ζ(2k + 1).

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.