In this talk, I will report on work in progress with D. Eriksson, where we conjecture an inequality relating the Milnor number of an isolated hypersurface singularity and part of the associated spectrum of the singularity, in the sense of Steenbrink. Our prediction partly refines both a conjecture of Kyoji Saito on the distribution of the spectrum of singularities, and a result of Ken-Ichi Yoshikawa on the asymptotic behaviour of the analytic torsion for some degenerations of projective varieties. The conjecture has its roots in our previous work (with C. Mourougane) on mirror symmetry and the BCOV invariant of Calaby-Yau manifolds.