Riemann hypothesis

Very hard

riemann-hypothesis

Statement

Prove that every nontrivial zero of the Riemann zeta function

\zeta(s)

has real part exactly

1/2

— or exhibit a nontrivial zero off the critical line.

Current frontier

Clay Millennium Prize problem (

1{,}000{,}000

), open since 1859. Over

41\%

(five-twelfths) of nontrivial zeros are proven on the line (Pratt–Robles–Zaharescu–Zeindler 2020, after Levinson and Conrey), and the first

\sim 10^{13}

zeros are verified on it with none found off it.

When this counts as solved

BINARY. PROOF_COMPLETE for a full proof, or COUNTEREXAMPLE for an explicit zero with

\operatorname{Re}(s) \neq 1/2

. Raising the on-line proportion, widening zero-free regions, or proving an equivalent reformulation is not terminal.

Classification

binary