Bulletin of the
Korean Mathematical Society BKMS

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