Nevanlinna exceptional-radius error notation

Assume the Axiom of Countable Choice (The Axiom of Countable Choice (${\mathrm{A}\mathrm{C}}_{\omega }$)). Under this assumption the Lebesgue measurable subsets of $\mathbb{R}$ form a $\sigma$-algebra and Lebesgue measure is a complete measure (Lebesgue measurable sets, the family $\mathcal{L}({\mathbb{R}}^n)$, and the restricted set function ${\lambda }_n$, Assuming countable choice, $\mathcal{L}({\mathbb{R}}^n)$ is a sigma-algebra containing every elementary set and ${\lambda }_n$ is a complete measure extending elementary volume).

Let $f$ be a nonconstant meromorphic function on $\mathbb{C}$, with characteristic $T(r,f)$ as in Counting, chordal proximity and characteristic. By Order and lower order from the Nevanlinna characteristic, $T(r,f)>1$ for all sufficiently large $r$.

An error term for $f$, written $S(r,f)$, is a function $e$ on a half-line $[r_0,\mathrm{\infty })$ with $r_0\ge 1$ for which there are a constant $C\ge 0$, a radius $r_0^{\mathrm{\prime }}\ge r_0$, and a Lebesgue measurable set $E\subseteq [r_0^{\mathrm{\prime }},\mathrm{\infty })$ of finite linear measure, $\lambda (E)<\mathrm{\infty }$, such that

$\mathrm{\mid }e(r)\mathrm{\mid }\le C({\log }^{+}T(r,f)+\log r)\text{for every }r\ge r_0^{\mathrm{\prime }}\text{ with }r\mathrm{\notin }E\mathrm{.}$

The notation $X(r,f)=S(r,f)$ means that the function $r\mapsto X(r,f)$ is an error term in this sense; the constant $C$, the threshold $r_0^{\mathrm{\prime }}$ and the exceptional set $E$ belong to that particular occurrence. No bound is asserted at the radii belonging to $E$.

• The exceptional set is part of each occurrence. Two occurrences of $S(r,f)$ in one formula may use different constants, thresholds and exceptional sets. A chain of estimates that uses $k$ occurrences may take the union $E_1\cup \cdots \cup E_k$ as a common exceptional set; a finite union of sets of finite linear measure again has finite linear measure, and the sum of the constants bounds the sum of the error terms.
• $S(r,f)$ denotes no single fixed function. Error terms for fixed $f$ are closed under finite real linear combinations on a common half-line, by the finite-union estimate above. The symbol abbreviates "some function satisfying the displayed bound"; replacing the constant or the exceptional set by larger ones produces another valid occurrence of the same symbol.
• No all-radius bound and no sharper order is implicit. Membership in $S(r,f)$ alone gives no information at exceptional radii and no information beyond the stated bound. The logarithmic-derivative lemma supplies an all-radius $O(\log r)$ estimate when $f$ has finite order and an all-radius $O(1)$ estimate when $f$ is rational. These are sufficient hypotheses for those estimates, not necessary ones: for $f(z)=e^z$, $f^{\mathrm{\prime }}\mathrm{/}f=1$ and $m_0(r,f^{\mathrm{\prime }}\mathrm{/}f)=0$ at every radius although $f$ is transcendental. No refinement of an arbitrary occurrence of $S(r,f)$ follows solely from the order or rationality of $f$.
• The exact use of Countable Choice. It is used only through the published measure interface: finite linear measure of $E$ and its finite unions, and the measurability of the sets of bad radii that occur. Every occurrence of $S(r,f)$ in this page carries the assumption explicitly, and no stronger choice principle is used.