  We generalize the "miraculous cancellation" formulas of Alvarez-Gaum\'e,
Witten and Kefeng Liu to a twisted version where an extra complex line bundle
is involved. We also apply our result to discuss intrinsic relations between
the higher dimensional Rokhlin type congruence formulas of Ochanine [O1],
Finashin [F] and Zhang [Z1, 2]. In particular, an analytic proof of the
Finashin congruence formula is given.
