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 A note on recurrence formula for values of the Euler zeta functions $\zeta_E (2n)$ at positive integers Bull. Korean Math. Soc. 2014 Vol. 51, No. 5, 1425-1432 https://doi.org/10.4134/BKMS.2014.51.5.1425Printed September 30, 2014 Hui Young Lee and Cheon Seoung Ryoo Hannam University, Hannam University Abstract : The Euler zeta function is defined by $\zeta_E(s)\!=\! \sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{n^s}$. The purpose of this paper is to find formulas of the Euler zeta function's values. In this paper, for $s\in \mathbb N$ we find the recurrence formula of $\zeta_E(2s)$ using the Fourier series. Also we find the recurrence formula of $\sum_{n=1}^{\infty} \frac{(-1)^{n-1}}{(2n-1)^{2s-1}}$, where $s\geq 2 (\in \mathbb N)$. Keywords : zeta function, Euler zeta function, Fourier series MSC numbers : Primary 42B05, 11B68, 11S40, 11S80 Downloads: Full-text PDF