Disclaimer: I had a hard time choosing the title.
Hello MSE!
So in this post I was wondering why $e^\gamma$ has importance in number theory while $\gamma$ and $e$ don't. But I have another question related to this. It is known that: $$e^\gamma=\lim_{k\rightarrow\infty}\left[\frac{1}{\ln p_k}\prod_{0<i\le k}\frac{p_i}{p_i-1}\right]$$But when we take the logarithm of both sides, and bring it into the limit (which is allowed since the logarithm is continuous), we get this limit for $\gamma$: $$\gamma=\lim_{k\rightarrow\infty}\left[-\ln\ln p_k+\sum_{0<i\le k}(\ln p_i-\ln(p_i-1))\right]$$What makes the limit for $e^\gamma$ more important than the limit for $\gamma$? I could tell that there is a product in the first limit while there is a sum in the second. Does this have to do with the numbers' importance in number theory?