  Determinants of invertible pseudo-differential operators (PDOs) close to
positive self-adjoint ones are defined throughthe zeta-function regularization.
We define a multiplicative anomaly as the ratio $\det(AB)/(\det(A)\det(B))$
considered as a functionon pairs of elliptic PDOs. We obtained an explicit
formula for the multiplicative anomaly in terms of symbols of operators. For a
certain natural classof PDOs on odd-dimensional manifolds generalizing the
class of ellipticdifferential operators, the multiplicative anomaly is
identically $1$. For elliptic PDOs from this class a holomorphic determinant
and a determinant for zero orders PDOs are introduced. Using various algebraic,
analytic, and topological tools we study local and global properties of the
multiplicative anomaly and of the determinant Lie group closely related with
it. The Lie algebra for the determinant Lie group has a description in terms of
symbols only. Our main discovery is that there is a {\em quadratic
non-linearity} hidden in the definition of determinants of PDOs through
zeta-functions. The natural explanation of this non-linearity follows from
complex-analytic properties of a new trace functional TR on PDOs of non-integer
orders. Using TR we easily reproduce known facts about noncommutative residues
of PDOs and obtain several new results. In particular, we describe a structure
of derivatives of zeta-functions at zero as of functions on logarithms of
elliptic PDOs. We propose several definitions extending zeta-regularized
determinants to general elliptic PDOs. For elliptic PDOs of nonzero complex
orders we introduce a canonical determinant in its natural domain of
definition.
