2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/229846We analyze aspects of the holographic principle relevant to the quantum gravity partition functions in Euclidean sector of AdS$_3$. The sum of the known contributions to the partitions functions can be presented exactly, including corrections, in the form where the Patterson-Selberg zeta function involves.8 pages, no figures. Some misprints have been corrected, title changed. To appear in the J. Phys. A (2009)High Energy Physics - Theorytext