  We compute partition functions of the deformed multiple M5-branes theory on
$K3\times T^2$ using the refined topological vertex formalism and the Borcherds
lift. The deformation is related to the mass deformation in the corresponding
four dimensional $N=4$ $SU(N)$ gauge theory on $K3$. The seed of the
  Borcherd-lift is calculated by taking the universal part of the type IIb
little string free energy of the CY3-fold $X_{N,1}$. We provide explicit
modular covariant expressions, as expansions in the mass parameter $m$, of the
genus two Siegel modular forms produced by the Borcherds lift of the first few
seed functions.
  We also discuss the relation between genus-one free energy and Ray-Singer
Torsion, and the automorphic properties of the latter.
