A note on the values of Euler zeta functions at posive integers (Q2907405)

Recently, the Euler zeta-function was introduced by \textit{T. Kim} [Abstr. Appl. Anal. 2008, Article ID 581582, 11 p. (2008; Zbl 1145.11019)] as NEWLINE\[NEWLINE\zeta_E(s)=2\sum_{n=1}^\infty \frac{(-1)^{n-1}}{n^s},\quad (s\in\mathbb C)NEWLINE\]NEWLINE In the paper under review the authors consider the values of the Euler zeta-function at positive integers. The corresponding values for \(\zeta_E(2)\), \(\zeta_E(4)\), and \(\zeta_E(6)\) are found and finally the formula NEWLINE\[NEWLINE\zeta_E(2n)=\frac{(-1)^n\pi^{2n}(2-4^n)}{2(2n-1)!(1-4^n)} E_{2n-1}NEWLINE\]NEWLINE is derived in analogy to the well known formula for \(\zeta(2n)\) in terms of the Bernoulli numbers \(B_{2n}\).