La relation entre $\zeta(4n-1)$, $\zeta(2p)$ et $\zeta(4n-1-2p)$

Abstract

The functional relation of the Riemann z\^eta function provides us with neither the nature nor the expression of z\^eta at positive odd numbers. From the function $F(z)=\frac{z^{-2n}}{e^z-1}$, we find a functional relation involving $\zeta(4n- 1)$, $\zeta(2p)$ and $\zeta(4n-1-2p)$. It is given by: \begin{equation} \zeta(4n-1)=\frac{1}{2n-1}\sum_{p=1}^{2n-2}\zeta(2p)\zeta(4n-1-2p). \end{equation} $n=2, 3, 4, 5, 6, ...$ From this formula we introduce a new approach to study the nature of $\zeta$ on these integers.