  The purpose of this article is to study the asymptotic expansion of
Ray-Singer analytic tosion associated with increasing powers p of a given
positive line bundle. Here we prove that the asymptotic expansion associated to
a manifold contains only the terms of the form $p^{n-i} \log p, p^{n-i}$ for
$i$-natural. For the two leading terms it was proved by Bismut and Vasserot in
1989. We will calculate the coefficients of the terms $p^{n-1} \log p, p^{n-1}$
in the Kahler case and thus answer the question posed in the recent work of
Klevtsov, Ma, Marinescu and Wiegmann about quantuum Hall effect. Our second
result concerns the general asymptotic expansion of Ray-Singer analytic torsion
for an orbifold.
