Assertion: riemann hypothesis | /r/HasProperty | true
Facets:
value: under the contrary assumption
type: OTHER
value: space of functions of the form
type: OTHER
Source sentences:
text: for all n > 5040 if and only if the Riemann hypothesis is true,
text: If the generalized Riemann hypothesis is false, then the theorem is true.
text: Nyman (1950) proved that the Riemann hypothesis is true if and only if the space of functions of the form
text: Salem (1953) showed that the Riemann hypothesis is true if and only if the integral equation
text: If the generalized Riemann hypothesis is true, then the theorem is true.
text: The method of proof is interesting, in that the inequality is shown first under the assumption that the Riemann hypothesis is true, secondly under the contrary assumption.
text: He showed that this in turn would imply that the Riemann hypothesis is true.
text: However one would expect this to happen occasionally just by chance even if the Riemann hypothesis were true, and Odlyzko's calculations suggest that nearby pairs of zeros occur just as often as predicted by Montgomery's conjecture.
text: Care should be taken to understand what is meant by saying the generalized Riemann hypothesis is false: one should specify exactly which class of Dirichlet series has a counterexample.