Riemann hypothesis

The zeta function encodes the primes, and the location of its zeros controls how far the count of primes up to x can stray from its smooth estimate. If the hypothesis holds, the error is about the square root of x, the best that could be hoped for. Over ten trillion zeros have been computed and all lie on the line. Hundreds of theorems are proved conditionally on it, so a proof would settle them all at once. The analogue for curves over finite fields was proved by Deligne in 1974, which is the strongest evidence that the statement is true and the main source of ideas for attacking it.

Posed 1859, Bernhard Riemann.

What would settle it

Lean statement in formal-conjectures

Prize

$1,000,000 Clay Millennium Prize

Related problems

• The original curation links this problem to Weak Goldbach conjecture; the argument for the link has not been written yet.
• The original curation links this problem to Weil conjectures; the argument for the link has not been written yet.

Sources

• Wikipedia: Riemann hypothesis
• Clay Mathematics Institute

Number theory · Mathematics