Infinite product $1+1/k^3$

Question

Ramanujan's notebooks contain the result $$\prod_{k=1}^{\infty} \Big( 1 + \frac{1}{k^3}\Big) = \frac{1}{\pi} \mathrm{cosh}\Big( \frac{\pi \sqrt{3}}{2}\Big).$$ It doesn't seem like this is proved there but I guess it is well-known. How could you show this?

Answer 1

Since the Weierstrass product for the Gamma function gives: $$\frac{1}{\Gamma(z)} = z\, e^{\gamma z}\cdot \prod_{k=1}^{+\infty}\left(1+\frac{z}{k}\right)e^{-\frac{z}{k}}$$ given that $\omega$ is a primitive third root of unity, we have: $$\prod_{k=1}^{+\infty}\left(1+\frac{z^3}{k^3}\right)=\frac{1}{z^3 \Gamma(z)\Gamma(\omega z)\Gamma(\omega^2 z)}=\frac{1}{\Gamma(z+1)\Gamma(\omega z+1)\Gamma(\omega^2z+1)},$$ hence: $$\prod_{k=1}^{+\infty}\left(1+\frac{1}{k^3}\right)=\frac{1}{\Gamma(\omega+1)\Gamma(\omega^2+1)},$$ but since $(\omega+1)+(\omega^2+1)=1$ and $$\Gamma(z)\Gamma(1-z)=\frac{\pi}{\sin(\pi z)},$$ it follows that: $$\prod_{k=1}^{+\infty}\left(1+\frac{1}{k^3}\right)=\frac{1}{\pi}\sin(\pi(\omega+1))=\frac{1}{\pi}\sin\left(\frac{\pi}{2}+i\frac{\pi\sqrt{3}}{2}\right)=\frac{1}{\pi}\cosh\frac{\pi\sqrt{3}}{2}.$$