These are notes of lectures on Nevanlinna theory, in the classical case
of meromorphic functions, and the generalization by Carlson-Griffith to
equidimensional holomorphic maps using as domain space finite coverings
of C resp. Cn. Conjecturally best possible error terms are
obtained following a method of Ahlfors and Wong. This is especially
significant when obtaining uniformity for the error term w.r.t.
coverings, since the analytic yields case a strong version of Vojta's
conjectures in the number-theoretic case involving the theory of
heights. The counting function for the ramified locus in the analytic
case is the analogue of the normalized logarithmetic discriminant in the
number-theoretic case, and is seen to occur with the expected
coefficient 1. The error terms are given involving an approximating
function (type function) similar to the probabilistic type function of
Khitchine in number theory. The leisurely exposition allows readers with
no background in Nevanlinna Theory to approach some of the basic
remaining problems around the error term. It may be used as a
continuation of a graduate course in complex analysis, also leading into
complex differential geometry.