Recurrence Relations for Values of the Riemann Zeta Function in Odd Integers

Abstract

It is commonly known that $\zeta (2k)=q_k\frac{\zeta (2k+2)}{{\pi }^2}$ with known rational numbers $q_k$. In this work we construct recurrence relations of the form ${\sum }_{k=1}^{\infty }{r}_k\frac{\zeta (2k+1)}{{\pi }^{2k}}=0$ and show that series representations for the coefficients $r_k\in \mathbb{R}$ can be computed explicitly.