The specific heat of single crystals of HgBa2Ca2Cu3O8+delta and HgBa2C
uO4+delta have been measured in fields up to 7 T-c by a sensitive ac t
echnique capable of measuring microgram samples. For both Hg-based sup
erconductors there is evidence of strong thermal fluctuations in the s
pecific heat with the mean-field step being replaced by a quasi-logari
thmic divergence of C at T-c in zero field. The almost symmetric shape
of the anomaly is in sharp contrast to that observed in YBa2Cu3O6.95,
and seems to be intrinsic to the quasi-two-dimensiona1 high-T-c super
conductors. In increasing field there is a strong reduction in the hei
ght of the anomaly with the peak position being only weakly shifted to
lower temperature. The applicability of fluctuation theories to the d
ata will be discussed.