Great question. The formula works whether or not the zeros have real part 0.5, but its implications are different. If you're counting primes up to x you will get the main term (roughly x/log x) and each zero p will give an oscillating error term ~ x^p/p. If the real part is always 0.5 then these error terms are all on sqrt(x) scale, which is what you would expect from random variation. If on the other hand you have a single zero right of the line, say at real part 3/4 (and mirror image one at real part 1/4) then now you get a secondary error term of size x^(3/4) that dominates all of the other error terms, giving some extra structure to the primes: they're no longer random, you can predict where they are more or less likely.
It's made abundantly clear: they learned to love it through the course they were fortunate enough to take as an undergraduate, and they learned to fear it later, as a researcher, when they realised that what there was to fear was wasting their career attacking something that was probably just too big and too hard to be sensible for them to attack.
Still, a number of people who've worked most closely with the Riemann zeta function, even the ones who led the computations and numerical verifications, have expressed doubts on whether the Riemann Hypothesis is actually true: https://arxiv.org/abs/math/0311162
Of course there is hope that a proof of RH would reveal some deeper understanding -- not merely a conformation.
Do you know if anyone has tabulated stats on conjecture proofs? That could be a fun dataset to examine.
The article talks about the personal story of how the author first met the Riemann Hypothesis.
Unless the theory also leads to faster factorization methods
If so much technology is built around a hypothesis, it is essentially building a large house of cards if the hypothesis turns out to be incorrect.
Imagine if the world set up a monetary system and what would happen after 100 years if suddenly everyone could print money when a fatal flaw in a system was discovered.
Discussions based on the title are useless. Good articles frequently have titles that are bad or not even relevant. Title of an article:
1. May not be chosen by the author. Its often editor decision.
2. Article can have multiple changing titles. For clickbait reasons.
Discussions based on title are generally worthless. In the HN you can ask someone to change the title to be more relevant.
But it fails on two counts. One is that it doesn't say what the title claims. We know now what the Rieman Hypothesis is, but how did the author learned to love, and more importantly to "fear" the RH? What's there to fear? The name of the famous movie (with the bomb) doesn't even have "fear" in it, so the author had something in mind with "fear", but after watching the video, I have no clue what.
Second is the promise in the beginning of the video to give us a hint why the RH is important. What we learn is that somehow each zeta zero adds another harmonic to some type of series approximation of the (cousin of the) prime counting function. Which is great; if we add enough harmonics we get to approximate this function as precisely as we want. But why is it important that these zeros are on the vertical line Im z = 0.5 ? I have absolutely no clue.
Don't get me wrong, I find that watching this video was a very good investment of 16 minutes of my life. But there is no need to overpromise, especially since the video delivers a lot as is.