This article says the implications of the newly proved quasi-Riemann hypothesis are "staggering" - but it doesn't mention any. So what are they? 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 |π(n) - li(n)| ≤ n / ln(n)¹⁰⁰⁰ 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 |π(n) - li(n)| ≤ C nᵖ 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 |π(n) - li(n)| ≤ C nᵖ for any p > 7/8. It also gave a shorter, clearer proof that |π(n) - li(n)| ≤ C nᵖ 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! (1/2) https://www.quantamagazine.org/ai-proved-the-quasi-riemann-hypothesis-what-happens-now-20261019/