The Riemann hypothesis
Is there a hidden order in the primes?
01 / The question
Primes are the building blocks of whole numbers. Their spacing is irregular, but their overall density has a pattern. The Riemann hypothesis concerns the nontrivial zeros of the zeta function and the precision with which we can describe prime distribution.
02 / Why it is hard
Checking any finite collection of zeros cannot establish a statement about all zeros. A convincing pattern needs an argument that covers every case.
03 / Progress & scope
Clay still lists the full hypothesis as unsolved. A result about a larger zero-free region would need to be compared with the exact critical-line claim; it would not automatically prove it.
OpenAI Math · GitHub
Family 003 · The quasi-Riemann hypothesis
The repository reports a zero-free half-plane result. This has a different scope from the full Riemann hypothesis.
Read the related researchTry an experiment
Make an abstract question tangible
The prime-counting experiment below illustrates average density. It does not compute zeta zeros or test the Riemann hypothesis.
Blue · exact prime count168
Dashed · n / ln(n) estimate144.8
The curves need not coincide. The prime number theorem concerns an asymptotic ratio; this finite sample proves neither that theorem nor the Riemann hypothesis.
Sources & further reading
Clay Mathematics InstituteSources last checked 2026-10-20. Status reflects an editorial review on that date, not continuous live monitoring.