Riemann hypothesis

Very hard

riemann-hypothesis

Statement

Prove that every nontrivial zero of the Riemann zeta function $ζ (s)$ 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 $\sim 1 0^{13}$ 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 $Re (s) \neq = 1/2$ . Raising the on-line proportion, widening zero-free regions, or proving an equivalent reformulation is not terminal.

Classification

Binary

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