  In this work, we employ the Hamiltonian approach to analyze the Lorentzian
path integral in 2+1 AdS gravity, with an aim to sum over all possible
geometries, including naked singularities and BTZ black holes, between fixed
initial and final surfaces. A novel path integral measure, grounded on the
metric over mini-superspace, is proposed for executing the path integral.
Shifting to Euclidean geometries, we derive the temperature and angular
potential of Euclidean naked singularities within the partition function.
Significantly, without utilizing conformal field theory, we extract logarithmic
corrections to the entropy of BTZ black holes and compute the entropy of naked
singularities, along with their log corrections.
