  We prove an asymptotic formula for the determinant of the bundle Laplacian on
discrete $d$-dimensional tori as the number of vertices tends to infinity. This
determinant has a combinatorial interpretation in terms of cycle-rooted
spanning forests. We also establish a relation (in the limit) between the
spectral zeta function of a line bundle over a discrete torus, the spectral
zeta function of the infinite graph $\mathbb{Z}^d$ and the Epstein-Hurwitz zeta
function. The latter can be viewed as the spectral zeta function of the twisted
continuous torus which is the limit of the sequence of discrete tori.
