Z coth(z) = 2 Sum of (-1)^(n+1) zeta(2n) z^(2n)/pi^(2n)

Theorem

The following formula holds for $|z|<\pi$: $$z \mathrm{coth}(z) = 2 \displaystyle\sum_{k=0}^{\infty} (-1)^{k+1} \zeta(2k) \dfrac{z^{2k}}{\pi^{2k}},$$ where $\mathrm{coth}$ denotes hyperbolic cotangent, $\zeta$ denotes Riemann zeta, and $\pi$ denotes pi.

References

• 1953: Arthur Erdélyi, Wilhelm Magnus, Fritz Oberhettinger and Francesco G. Tricomi: Higher Transcendental Functions Volume I ... (previous) ... (next): $\S 1.20 (2)$