  This paper studies algebraic and analytic structures associated with the
Lerch zeta function, extending the complex variables viewpoint taken in part
II. The Lerch transcendent $\Phi(s, z, c)$ is obtained from the Lerch zeta
function $\zeta(s, a, c)$ by the change of variable $z=e^{2 \pi i a}$. We show
that it analytically continues to a maximal domain of holomorphy in three
complex variables $(s, z, c)$ as a multivalued function defined over the base
manifold ${\bf C} \times P^1({\bf C} \smallsetminus \{0, 1, \infty\}) \times
({\bf C}\smallsetminus {\bf Z})$. and compute the monodromy functions defining
the multivaluedness. For positive integer values s=m and c=1 this function is
closely related to the classical m-th order polylogarithm $Li_m(z)$ We study
its behavior as a function of two variables $(z, c)$ for special values where
s=m is an integer. For $m \ge 1$ it gives a one-parameter deformation of the
polylogarithm, and satisfies a linear ODE with coefficients depending on c, of
order m+1 of Fuchsian type. We determine its (m+1)-dimensional monodromy
representation, which is a non-algebraic deformation of the monodromy of
$Li_m(z).$
