  In this note, making use of noncommutative $l$-adic cohomology, we extend the
generalized Riemann hypothesis from the realm of algebraic geometry to the
broad setting of geometric noncommutative schemes in the sense of Orlov. As a
first application, we prove that the generalized Riemann hypothesis is
invariant under derived equivalences and homological projective duality. As a
second application, we prove the noncommutative generalized Riemann hypothesis
in some new cases.
