We can't prove the primes hide no secret pattern

The problem

Prime numbers look scattered almost at random, yet in 1859 Bernhard Riemann found a hidden machine behind them: a function whose zeros control exactly how the primes fluctuate. He conjectured that every one of the meaningful zeros sits precisely on a single line — and if that's true, the primes are as orderly as they could possibly be. Computers have checked the first ten trillion zeros and every single one lands exactly on the line, but ten trillion confirmations are not a proof. It is widely regarded as the deepest open problem in mathematics, having survived 167 years of assault by many of history's greatest mathematicians.

Why it matters

Hundreds of theorems across number theory are proved 'assuming RH' — a proof would instantly cement them all, while a single off-line zero would demolish them. Our finest estimates of how primes are distributed, and the confidence behind prime-based cryptography, lean on understanding these zeros.

Progress so far

• 1914G. H. Hardy proves infinitely many zeros lie on the critical line
• 1974Norman Levinson shows at least one-third of the zeros are on the line
• 1989Brian Conrey raises that to more than two-fifths
• 2004Xavier Gourdon verifies the first ten trillion zeros computationally; all lie on the line

References

1. Clay Mathematics Institute — official Millennium Prize problem page; status: unsolved
2. Wikipedia — full history, partial results, and numerical evidence