  In this article, following an insight of Kontsevich, we extend the famous
Weil conjecture (as well as the strong form of the Tate conjecture) from the
realm of algebraic geometry to the broad noncommutative setting of dg
categories. As a first application, we prove the noncommutative Weil conjecture
(and the noncommutative strong form of the Tate conjecture) in the following
cases: twisted schemes, Calabi-Yau dg categories associated to hypersurfaces,
noncommutative gluings of schemes, root stacks, (twisted) global orbifolds,
connective dg algebras, and finite-dimensional dg algebras. As a second
application, we provide an alternative noncommutative proof of Weil's original
conjecture (which avoids the involved tools used by Deligne) in the cases of
intersections of two quadrics and linear sections of determinantal varieties.
Finally, we extend also the classical theory of L-functions (as well as the
corresponding conjectures of Tate and Beilinson) from the realm of algebraic
geometry to the broad noncommutative setting of dg categories. Among other
applications, this leads to an alternative noncommutative proof of a celebrated
convergence result of Serre.
