  By means of an identity that equates elliptic genus partition function of a
supersymmetric sigma model on the $N$-fold symmetric product $S^N X$ of $X$
($S^N X=X^N/S_N$, $S_N$ is the symmetric group of $N$ elements) to the
partition function of a second quantized string theory, we derive the
asymptotic expansion of the partition function as well as the asymptotic for
the degeneracy of spectrum in string theory. The asymptotic expansion for the
state counting reproduces the logarithmic correction to the black hole entropy.
