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.2014.51.5.1425
Printed 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
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