  We obtain asymptotic lower bounds for the spectral function of the Laplacian
and for the remainder in local Weyl's law on manifolds. In the negatively
curved case, thermodynamic formalism is applied to improve the estimates. Key
ingredients of the proof include the wave equation parametrix, a pretrace
formula and the Dirichlet box principle. Our results develop and extend the
unpublished thesis of A. Karnaukh.
