  In this paper, we introduce the notion of spectral genus $\widetilde{p}_{g}$
of a germ of an isolated hypersurface singularity $(\mathbb{C}^{n+1}, 0) \to
(\mathbb{C}, 0)$, defined as a sum of small exponents of monodromy eigenvalues.
The number of these is equal to the geometric genus $p_{g}$, and hence
$\widetilde{p}_g$ can be considered as a secondary invariant to it. We then
explore a secondary version of the Durfee conjecture on $p_{g}$, and we predict
an inequality between $\widetilde{p}_{g}$ and the Milnor number $\mu$, to the
effect that $$\widetilde{p}_g\leq\frac{\mu-1}{(n+2)!}.$$ We provide evidence by
confirming our conjecture in several cases, including homogeneous singularities
and singularities with large Newton polyhedra, and quasi-homogeneous or
irreducible curve singularities. We also show that a weaker inequality follows
from Durfee's conjecture, and hence holds for quasi-homogeneous singularities
and curve singularities.
  Our conjecture is shown to relate closely to the asymptotic behavior of the
holomorphic analytic torsion of the sheaf of holomorphic functions on a
degeneration of projective varieties, potentially indicating deeper geometric
and analytic connections.
