OpenAI just posted a paper claiming to prove the quasi-Riemann hypothesis: no zero of the Riemann zeta function, or of any Dirichlet L-function, has real part above 11/12. A companion paper pushes it to 7/8.
Since Riemann's 1859 paper, nobody had proven any fixed bound like this. If it holds up, it also rules out Siegel zeros.
What this means:
The zeta function encodes how the prime numbers are distributed. Its inputs are complex numbers, so each zero has a real part and an imaginary part.
There are some trivial zeros, we dont care about them. Every other zero is known to have real part between 0 and 1. The Riemann Hypothesis says they should all have real part exactly 1/2.
Nobody has proven that. The best known bounds got closer and closer to 1 as the imaginary part grew. That weaker statement is the quasi-Riemann hypothesis, and it's what this paper claims.
Siegel zeros are hypothetical zeros extremely close to 1 that have haunted number theory for nearly a century. This result would rule them out.
Its still under experts review, but its huge!
We’re releasing a broad range of new mathematical results produced by an internal frontier model.
We’ve been consulting with the independent Advisory Group on Mathematics and Artificial Intelligence at the Institute for Advanced Study, and we have drawn on their advice and public recommendations to inform how we release these results.
github.com/openai/math