continuum hypothesis | Riemann hypothesis

The continuum hypothesis states that there is no infinite set with a cardinal number between that of the "small" infinite set of integers and the "large" infinite set of real numbers.

The Riemann hypothesis posits that the nontrivial zeros of the Riemann zeta function ζ(s) all lie on the critical line Re(s) = 1/2.

| solution continuum hypothesis | undecidable

| formal statement Riemann hypothesis | for all _(n, n element Z ∧ n!=0)Re(ρ_n) = 1/2

| continuum hypothesis | Riemann hypothesis formulation date | 1877 (148 years ago) | 1859 (166 years ago) formulators | Georg Cantor | Bernhard Riemann status | ambiguous | open proof date | 1963 (86 years later) (62 years ago) | provers | Kurt Gödel | Paul Joseph Cohen | additional people involved | David Hilbert |

Re(ρ_n) = 1/2

Proved by Gödel and Cohen to be undecidable within Zermelo-Frankel set theory with or without the axiom of choice, but there is no consensus on whether this is a solution to the problem.

It has been verified that the first 1×10^13 nontrivial zeros of the zeta function lie on the critical line. Conrey (1989) proved that at least 40% of the nontrivial zeros of the zeta function lie on the critical line.

| prizes offered for solution | prizes awarded for solution continuum hypothesis | | Paul Cohen received the Fields Medal in 1966 for showing that if set theory is consistent, then no contradiction would arise if the negation of the continuum hypothesis was added to set theory. Riemann hypothesis | $1 million "Millennium Prize Problem" of The Clay Mathematics Institute. |

mathematical hypotheses