Nevanlinna deficiency (${\delta }_f(a)=1-\mathrm{limsup}N(r;a)/T_f(r)$)

For a nonconstant meromorphic function, its deficiency at a point of the Riemann sphere is the displayed asymptotic proportion missing from the full Nevanlinna integrated counting function. The Nevanlinna first main theorem bounds $N(r;a)$ by $T_f(r)+O(1)$ and identifies deficiency with the lower limiting normalized proximity. An omitted value has deficiency one.

The rational composition law for the Nevanlinna characteristic gives $T_f(r)=3r/\pi +O(1)$. Its zeros are precisely $2\pi i\mathbb{Z}$, all of multiplicity two, so $N(r;0)=2r/\pi +O(\log r)$ and $\overline{N}(r;0)=r/\pi +O(\log r)$. Hence both its Nevanlinna deficiency and Nevanlinna ramification index at zero equal one third.