Euler's constant's role in the Prime numbers

Question

I know that: $$\lim_{k\rightarrow\infty}\frac{1}{\log p_n}\prod_{i=1}^k\frac{p_i}{p_i-1}=e^\gamma$$Where $p_n$ is the $n$th prime and $\gamma$ is Euler's constant. But why does this make the number $\gamma$ important in number theory (according to wikipedia)? I could say that $$\lim_{k\rightarrow\infty}H(k)-\frac{k}{\pi(k)}=\gamma$$Where $\pi(k)$ is the prime counting function (to prove this use the fact that $\pi(x)\sim\dfrac{x}{\log x}$ and then solve for $\log x$). But this doesn't make it useful.

Answer 1

The number $e^{\gamma}$ plays an important role in the theory of $y$-rough numbers , where the Buchstab-function can be used to estimate the number of $y$-rough numbers below $x$. And this function tends quickly to $e^{-\gamma}$

It is also useful to estimate the probability that a large random number $x$ is prime if it has no prime factor below $y$ , assuming $y$ itself is large , but $x$ much larger.

Neither the Euler number $e$ nor the Euler Mascheroni constant $\gamma$ are however important for all this, just the combined number.