  We study the implications of duality symmetry on the analyticity properties
of the partition function as it depends upon the compactification length. In
order to obtain non-trivial compactifications, we give a physical prescription
to get the Helmholtz free energy for any heterotic string supersymmetric or
not. After proving that the free energy is always invariant under the duality
transformation $R\rightarrow \alpha^{'}/(4R)$ and getting the zero temperature
theory whose partition function corresponds to the Helmholtz potential, we show
that the self-dual point $R_{0}=\sqrt{\alpha^{'}}/2$ is a generic singularity
as the Hagedorn one. The main difference between these two critical
compactification radii is that the term producing the singularity at the
self-dual point is finite for any $R \neq R_{0}$. We see that this behavior at
$R_{0}$ actually implies a loss of degrees of freedom below that point.
