On Nevanlinna's Secondary Deficiency

Analogies between the Nevanlinna theory and the theory of heights in number theory have motivated to the determination the precise error term of the second fundamental theorem of the Nevanlinna theory. The Nevanlinna secondary deeciency, introduced by P.M. Wong, gives the error term in a certain sense. We rst prove that, from a topological point of view, almost all meromorphic functions have secondary deeciency negative innnity and that the set of meromorphic functions having maximum secondary deeciency is dense in a standard topology. Then we give an improved upper bound on the secondary deeciency for meromorphic functions of nite order and show that the upper bound is sharp.