This question is for a very cool friend of mine. See, he really is interested on how seemingly separate concepts can be connected in such nice ways. He told me that he was losing his love for mathematics due to some personal stuff in his life. (I will NOT discuss his personal life, but it has nothing to do with me).
To make him feel better about math (or love of math for that matter), I was planning on giving him a sheet of paper with quite a few relations of $e$ and $\pi$
The ones I were going to give him were:
and how $e$ and $\pi$ both occur in the distribution of primes (bounds on primes has to do with $\ln$ and the regularization of the product of primes are $4\pi^2$)
Can I have a few more examples of any relations please? I feel it could mean a lot to him. I'm sorry if this is too soft for this site, but I didn't quite know where to ask it.