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We extend the calculus of adiabatic pseudo-differential operators to studythe adiabatic limit behavior of the eta and zeta functions of a differentialoperator $\delta$, constructed from an elliptic family of operators indexed by$S^1$. We show that the regularized values ${\eta}(\delta_t,0)$ and$t{\zeta}(\delta_t,0)$ are smooth functions of $t$ at $t=0$, and we identifytheir values at $t=0$ with the holonomy of the determinant bundle, respectivelywith a residue trace. For invertible families of operators, the functions${\eta}(\delta_t,s)$ and $t{\zeta}(\delta_t,s)$ are shown to extend smoothly to$t=0$ for all values of $s$. After normalizing with a Gamma factor, the zetafunction satisfies in the adiabatic limit an identity reminiscent of theRiemann zeta function, while the eta function converges to the volume of theBismut-Freed meromorphic family of connection 1-forms.Comment: 32 pages, final versio

Adiabatic limits of eta and zeta functions of elliptic operators