We study the degenerations of intermediate Jacobians by means of log geometry. We extend the family of intermediate Jacobians over a punctured disc to a "log intermediate Jacobian'' over a disc.

We present a formula describing modularity gap for Eisenstein series, which is written in terms of a certain double series. Limit values of the gap at nonzero rational points are expressible by Hurwitz zeta values. Our gap estimates near the origin are applied to examining the asymptotic behaviour of Ramanujan q-series and q-zeta values near the natural boundary |q|=1.

Let k be an integer with $k \ge 2$. We show that if G be a graph such that $|G| > 4k+1 -4\sqrt {k-1}$ and $bind(G)> {(2k-1)(n-1) \over k(n-2)},$ then G is a fractional k-deleted graph. We also show that in the case where k is even, if G be a graph such that $|G| > 4k+1 -4\sqrt {k}$ and $bind(G)> {(2k-1)(n-1) \over k(n-2)+1},$ then G is a fractional k-deleted graph.