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The first quantum corrections to the entropy for an eternal four-dimensional extremal Reissner-Nordstr\"om black hole is investigated at the one-loop level, in the large mass limit of the black hole, making use of the conformal techniques related to the optical metric. A leading exponential horizon divergence is found and other divergences appear due to the singular nature of the optical manifold. The area law is shown to be violated.
Cognola et al. (Sun,) studied this question.