Path Integrals, asymptotic expansions and Zeta determinants

It is "well-known" in quantum mechanics that the values of certain path integrals are given by associated zeta-determinants "up to a multiplicative constant“, by what is usually meant that one can only calculate the ratio of path integrals by the ratio of zeta functions. We investigate path integral formulas for the heat kernel of a Riemannian manifold and give a rigorous meaning to the above statements in this case. Furthermore, we show how the heat kernel asymptotics are related to formal asymptotic expansions on path space.