Open Question
No proof or disproof has been found since Riemann first conjectured it in 1859, despite it being one of the most heavily attacked problems in mathematics; computation has verified the first many trillions of zeros with no exception, which is strong evidence but not a proof.
What would resolve this A general proof (or a single confirmed counterexample, a nontrivial zero found off the critical line) covering every zero, not merely the ones checked so far; a Clay Mathematics Institute Millennium Prize of one million dollars is offered for a correct resolution either way.
OpenAnalytic number theoryClay Mathematics Institute
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