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