This is pure math, so not much, at least directly or immediately.
But I find it amusing how this argument always comes up and how it goes back millenia.
A student of Plato (428 B.C. -- 348 B.C.) once asked the great master, "What practical uses do these theorems serve? What is to be gained from them?" Plato's answer was immediate and peremptory. He turned to one of his slaves and said, "Give this young man an obol [a small Greek coin] so that he may feel that he has gained something from my teachings. Then expel him."
I didn’t mean to imply that it needed to be useful. I just hear about the hypothesis a lot and I wonder what the immediate knock-on effects would be. Does it unlock other theoretical work, are there other proofs that work or don’t work depending on it, etc. And if it has an effect on something real or tangible that’s even better.
These things are kind of like landing on the moon. The act of actually walking on the moon is fairly symbolic and ultimately the more important things were the achievements required to get there.
What the current state of the problem demonstrates is there's some unknown quantity of cutting edge work required that has yet to be done; maybe there's a cunning conceptual error in some field that has somehow missed everyone's eye or maybe there's an entirely new mathematical concept or tool which makes the problem straight-forward, but nobody has "invented it" yet.
One day, the time we are in right now will be "150 years ago". Contemporaries of any time think all fundamental things have been exhausted and discovered for whatever reason. It was as true in 1874 as 2024.
And they've always been wrong. Doesn't mean you or I can get there. It'll take some brilliant individual or team years of sweat equity, skill and profound luck but one day, right now too will be 150 years ago and people will look back on us as we look back on 1874 mathematics.
There's another possibility: that everything discoverable by humans as been discovered.
Before long, AIs and organizations will be claiming to prove things that no human has the capacity to verify with reasonable confidence, or humans will run or of capacity or make progress. We just had the 1000 page Geometric Langlands proof.
"everything discoverable by humans as been discovered" again, this claim is made by every generation like clockwork, going back as long as humans have been making claims that can be recorded.
In principle a proof of the Riemann Hypothesis could give you information about the distribution of primes and could possible make it easier to test for primality, in the worst case breaking modern encryption.
My apologies, that is not what I meant. I meant that the proof might break encryption. Again, might. As response to a list of possible real-world implications as the reader asked for.
But I find it amusing how this argument always comes up and how it goes back millenia.