I am an undergraduate student writing an expository thesis on the complex-analytic proof of the Prime Number Theorem.

I understand that applying the Mellin Transform to the partial sum of the van Mangoldt function Λ(x) makes it complex-analytic and related to zeta, so we can study the result's asymptotic behaviors and then take the Mellin Inversion ride back to the sum of primes. This makes sense to me on a surface level.

On a deeper level though, I know that the Mellin Transform is a tool for diagonalizing dilations (owing to its kernel being an eigenfunction for dilation). So there seems to be some deeper purpose to applying the Mellin Transform to Λ(x) and π(x), considering that the primes are a multiplicative idea. Perhaps we are "decomposing" prime distribution into some dilative frequency?

I think it would make more immediate intuitive sense to me if applying the Mellin Transform to a function representative of the group of positive integers gave prime distribution, considering that I conceptualize the primes as some kind of multiplicative bases, but it is confusing to me that in fact we should be going the opposite way.

I've been wracking my head with this problem for a few weeks and also have read up on everything I could on the Mellin Transform and the PNT, but no search of mine turned up yet an exact answer to this particular confusion.

If anyone could provide any tips/directions/hints or reading recommendations that might reveal some fundamental intuition about why it is valid to apply the Mellin Transform to prime distribution, I'd really appreciate it.