How can I use Mathematica to resolve what's missing in the connection between the prime number theorem and the MRB constant?

I've found the Wolfram Notebook assistant very useful when I'm too lazy to get to the bottom of a proof for an idea that I have, or even perfecting the concept.

Here is one I'm working on.
In essence, the MRB constant is the upper limit of the partial sums of n^(1/n).

n^(1/n)=Exp[Log[n]/n]

For large values of n, the ratio PrimePi(n)/(n/log n) approaches 1. This relationship is a result of the Prime Number Theorem.
Since 1/(n/Log[n])=Log[n]/n, for a long time, I've been looking for a connection between the prime number theorem and the MRB constant.
You might have another take on it.