Consequences of progress toward the Riemann Hypothesis

The Riemann Hypothesis (RH) is the conjecture that all the zeros of the Riemann zeta function ζ(s) in the critical strip, i.e. the region of the complex plane with real part between 0 and 1, have real part equal to ½.

The Quasi Riemann Hypothesis (QRH) says that there exists a constant θ < 1 such that no zeros of ζ(s) have real part greater than θ. OpenAI has published a paper claiming QRH with θ = 7/8.

The RH is so important to number theory that even partial results can have big consequences. This post will focus on one consequence: the error term in the Prime Number Theorem.

The Prime Number Theorem says that π(x), the number of primes less than x, is asymptotically equal to Li(x). We’d like to know more specifically at what rate π(x) approaches Li(x).

The best known result before the QRH announcement was

If the QRH holds for some θ, such as OpenAI’s assertion that θ = 7/8,

Incidentally, you may have seen the Prime Number Theorem stated with x/log(x) rather than Li(x). These two functions are asymptotically equal, so they give the same theorem, if you’re not interested in quantifying the rate of convergence. The function Li(x) gives better error bounds.