This refinement of the Hilbert–Pólya conjecture is known as the "Berry conjecture" (or the "Berry–Keating conjecture"). As of 2008, it is still quite far from being concrete, as it is not clear on which space this operator should act in order to get the correct dynamics, nor how to regularize it in order to get the expected logarithmic corrections. Berry and Keating have conjectured that since this operator is invariant under dilations perhaps the boundary condition "f"("nx") = "f"("x") for integer "n" may help to get the correct asymptotic results valid for large "n"