First, remember the number of primes less than n is roughly n / ln(n). The Riemann hypothesis lets us make this more precise. So does the quasi-Riemann hypothesis, though it's not as good. Let's get into it!
For all this it's better to use, not n / ln(n), but a more precise estimate of the number of primes less than n: a function called li(n), which is the integral of 1/ln(x) from 0 to n. The difference between this and n / ln(n) is a lot bigger than the errors I'm going to talk about.
Before the recent work, we knew things like this: if π(n) is the number of primes less than n, then
if n is large enough. Indeed, we could replace the number 1000 by any other large number.
The Riemann hypothesis is equivalent to this: if π(n) is the number of primes less than n, then
for any p > 1/2, and some C that depends on p. We know we can't get p all the way down to 1/2.
So crudely, the Riemann hypothesis brings the error bound down from "a lot less than n, but still bigger than any smaller power of n" to "not much more than √n".
The quasi-Riemann hypothesis does the same thing but replacing √n with some bigger power of n. OpenAI gave a long, complicated proof that
for any p > 7/8. It also gave a shorter, clearer proof that
for any p > 11/12. Nobody has checked the first proof, but a lot of number theorists are checking the second one and it seems to be good!
Since 1897, we've known all the nontrivial zeros of the Riemann zeta function lie in this strip in the complex plane:
The quasi-Riemann hypothesis says they lie in some narrower strip
and right now we can probably choose a = 1/8, and definitely a = 1/12. The Riemann hypothesis claims all these zeros lie exactly on the line
The closer to this line we can show these zeros are, the more regular the distribution of primes must be!
But the quasi-Riemann hypothesis has other consequences, too — especially combined with the recent proof that a bunch of L-functions have no "Siegel zeros". I'll just mention one of the cuter, less technical results.
"Goldbach conjecture on average" now has a precise answer! You can count, for each even number up to n, the number of ways to write it as a sum of two primes, and add these counts up. The total is close to a simple formula, and the question is how close. The quasi-Riemann hypothesis puts a bound on the error!
Conversely, any bound of a certain type implies the quasi-Riemann hypothesis. But the quasi-Riemann hypothesis got there first!