Study of an infinite product

During some research, I obtained the following convergent product ${P}_{a}(x):=\prod _{j=1}^{\mathrm{\infty}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\left(\frac{x}{{j}^{a}}\right)\phantom{\rule{1em}{0ex}}(x\in \mathbb{R},a>1).$.

Considering how I got it, I know it's convergent and continuous at 0 (for any fixed a), but if I look at ${P}_{a}$ now, it doesn't seem so obvious for me.

Try: I showed that ${P}_{a}\in {L}^{1}(\mathbb{R})$, i.e. it is absolutely integrable on $\mathbb{R}$. Indeed, by using the linearization of the cosine function and the inequalities $\mathrm{ln}(1-y)\u2a7d-y$ (for any $y<1$) and $1-\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}z\u2a7e{z}^{2}/2$ (for any real z), we obtain

$\begin{array}{rcl}{P}_{a}(x{)}^{2}& =& \prod _{j=1}^{\mathrm{\infty}}(1-\frac{1-\mathrm{cos}(x\phantom{\rule{thinmathspace}{0ex}}{j}^{-a})}{2})\u2a7d\prod _{j>|x{|}^{1/a}}(1-\frac{1-\mathrm{cos}(x\phantom{\rule{thinmathspace}{0ex}}{j}^{-a})}{2})\u2a7d\mathrm{exp}(-C|x{|}^{1/a}),\end{array}$ for some absolute constant $C>0$. However, I have not been able to use this upper bound to prove continuity at 0 (via uniform convergence for example, if we can).

Question : I wanted to know how to study ${P}_{a}$ (e.g. its convergence and continuity at 0), if you think it has an other form "without product" (or other nice properties) and finally if anyone has already seen this type of product (in some references/articles), please.

During some research, I obtained the following convergent product ${P}_{a}(x):=\prod _{j=1}^{\mathrm{\infty}}\phantom{\rule{thinmathspace}{0ex}}\mathrm{cos}\left(\frac{x}{{j}^{a}}\right)\phantom{\rule{1em}{0ex}}(x\in \mathbb{R},a>1).$.

Considering how I got it, I know it's convergent and continuous at 0 (for any fixed a), but if I look at ${P}_{a}$ now, it doesn't seem so obvious for me.

Try: I showed that ${P}_{a}\in {L}^{1}(\mathbb{R})$, i.e. it is absolutely integrable on $\mathbb{R}$. Indeed, by using the linearization of the cosine function and the inequalities $\mathrm{ln}(1-y)\u2a7d-y$ (for any $y<1$) and $1-\mathrm{cos}\phantom{\rule{thinmathspace}{0ex}}z\u2a7e{z}^{2}/2$ (for any real z), we obtain

$\begin{array}{rcl}{P}_{a}(x{)}^{2}& =& \prod _{j=1}^{\mathrm{\infty}}(1-\frac{1-\mathrm{cos}(x\phantom{\rule{thinmathspace}{0ex}}{j}^{-a})}{2})\u2a7d\prod _{j>|x{|}^{1/a}}(1-\frac{1-\mathrm{cos}(x\phantom{\rule{thinmathspace}{0ex}}{j}^{-a})}{2})\u2a7d\mathrm{exp}(-C|x{|}^{1/a}),\end{array}$ for some absolute constant $C>0$. However, I have not been able to use this upper bound to prove continuity at 0 (via uniform convergence for example, if we can).

Question : I wanted to know how to study ${P}_{a}$ (e.g. its convergence and continuity at 0), if you think it has an other form "without product" (or other nice properties) and finally if anyone has already seen this type of product (in some references/articles), please.