For what values of $a$ does $P=\prod\limits_{k=1}^n a|\sin{k}|\to\infty$ as $n\to\infty$ ?
Experimenting on desmos, it seemed that if $a>2$ then $P\to\infty$, but some strange cases like $\prod\limits_{k=1}^{120000} 2.0001|\sin{k}|\approx 4\times10^{-17}$ made me doubt it.
Either there exists a critical value for $a$ such that $P\to\infty$, or $P\not\to\infty$ for all $a$. Either way, I think it's astounding.
Best Answer
This is only an approach of the problem. We have $P_n = a^n u_n$ where $u_n = \prod_{k=1}^n |\sin k|$. Clearly, $u_n$ converges to $0$, hence $\log u_n$ tends to $-\infty$.
Proposition Let $\ell = \limsup \frac{\log |\log u_n|}{\log n}$. If $\ell > 1$ then no such $a$ exists.
Proof: if $\ell > 1$, let $1 <\alpha< \ell$. For an infinite number of $n$ we have $|\log u_n| \ge n^\alpha$, hence $\log u_n\le -n^\alpha$, hence $\log P_n\le n \log a - n^\alpha$ which tends to $-\infty$ for this subsequence of ns.
Now have you tried this $\limsup$ numerically?
Second question Can we expect that for large $n$ \begin{equation} \frac{1}{n}\sum_{k=1}^n \log|\sin k| \approx \frac{1}{\pi}\int_0^\pi\log|\sin\theta| d \theta\quad? \end{equation}