  Let (M^n, g) be a closed smooth Riemannian spin manifold and denote by D its
Atiyah-Singer-Dirac operator. We study the variation of Riemannian metrics for
the zeta function and functional determinant of D^2, and prove finiteness of
the Morse index at stationary metrics, and local extremality at such metrics
under general, i.e. not only conformal, change of metrics.
  In even dimensions, which is also a new case for the conformal Laplacian, the
relevant stability operator is of log-polyhomogeneous pseudodifferential type,
and we prove new results of independent interest, on the spectrum for such
operators. We use this to prove local extremality under variation of the
Riemannian metric, which in the important example when (M^n, g) is the round
n-sphere, gives a partial verification of Branson's conjecture on the pattern
of extremals. Thus det(D^2) has a local (max, max, min, min) when the dimension
is (4k, 4k + 1, 4k + 2, 4k + 3), respectively.
