Yesterday, I encountered the product $\prod_{k=0}^{\infty}\cos(\frac{\pi}{2^{k+2}})$ in a post which is unfortunately deleted by its poster after some negative comments.
I found the product in the following way: Let $A_n=\prod_{k=0}^{n}\cos(\frac{\pi}{2^{k+2}})$ and $B_n=\prod_{k=0}^{n}\sin(\frac{\pi}{2^{k+2}})$. Then $A_nB_n=\frac{1}{2^{n+1}}B_{n-1}$ and we have $A_n=\frac{2}{\pi}\frac{\frac{\pi}{2^{n+2}}}{\sin(\frac{\pi}{2^{k+2}})}$. Thus, $\prod_{k=0}^{\infty}\cos(\frac{\pi}{2^{k+2}})=\lim_{n\rightarrow\infty}A_n=\frac{2}{\pi}.$
Then I thought about the infinite product $$\prod_{k=0}^{\infty}\cos(\frac{\pi}{p^{k+1}})=\cos(\frac{\pi}{p})\cos(\frac{\pi}{p^{2}})\cos(\frac{\pi}{p^{3}})...$$ where $p$ is an odd prime number. I am stuck.
Any comments or answers are wellcome.
Computing $$P_n=\prod_{k=0}^\infty \cos\left(\frac \pi {p^{k+1} } \right)\quad \implies \quad \log(P_n)=\sum_{k=0}^\infty\log\left( \cos\left(\frac \pi {p^{k+1} }\right) \right)$$ is not the most pleasant to compute.
We can use $$\log\left( \cos\left(\frac \pi {x }\right) \right)=\sum_{n=1}^\infty (-1)^n \frac{2^{2 n-3} \pi ^{2 n} (E_{2 n-1}(1)-E_{2 n-1}(0))}{n \,(2 n-1)!\, x^{2n}}$$ whcih gives
$$\log(P_n)=-\frac {\pi^2}{2(p^2-1)}-\frac {\pi^4}{12(p^4-1)}-\frac {\pi^6}{45(p^6-1)}-\frac{17 \pi ^8}{2520 \left(p^8-1\right)}-\frac{31 \pi ^{10}}{14175 \left(p^{10}-1\right)}-\cdots$$
Computing for a few integer values of $p$ (only the terms given above were used)
$$\left( \begin{array}{ccc} p & \text{estimate} & \text{solution} \\ 3 &\color{red}{ -0.76}0890548037401 & -0.762980593156087 \\ 4 &\color{red}{-0.3672}09038523256 & -0.367260902722768 \\ 5 &\color{red}{ -0.2201}77685381150 & -0.220180927677961 \\ 6 &\color{red}{ -0.147762}045679447 & -0.147762392383186 \\ 7 &\color{red}{ -0.106384}062181141 & -0.106384115219117 \\ 8 & \color{red}{-0.0803979}74954449 & -0.080397985451577 \\ 9 & \color{red}{-0.06296418}6060146 & -0.062964188584209 \\ 10 &\color{red}{ -0.050680335}287169 & -0.050680335994092 \\ 11 &\color{red}{ -0.041690186}591224 & -0.041690186815090 \\ 12 & \color{red}{-0.034907897}472455 & -0.034907897550889 \\ 13 & \color{red}{-0.029662552}271415 & -0.029662552301324 \\ 14 & \color{red}{-0.0255208681}33972 & -0.025520868146228 \\ 15 & \color{red}{-0.02219261524}3287 & -0.022192615248630 \\ 16 & \color{red}{-0.01947731794}2376 & -0.019477317944835 \\ 17 & \color{red}{-0.01723281562}4760 & -0.017232815625946 \\ 18 & \color{red}{-0.015355986601}374 & -0.015355986601970 \\ 19 &\color{red}{-0.01377053020}6716 & -0.013770530207027 \\ 20 & \color{red}{-0.012418995869}403 & -0.012418995869571 \end{array} \right)$$