---

Statement

Setting (p. 28). The paper assumes the standard notation of Nevanlinna theory: for a function $f$ meromorphic in $\mathbb C$, $n(r,a)$, also written $n(r,a,f)$, is the number of $a$-points of $f$ in $\{z:\lvert z\rvert<r\}$, counted with multiplicity. $A(r,f)$ is the mean number of sheets of the Riemann surface $F_r$ onto which $f$ maps $\{z:\lvert z\rvert<r\}$; the paper recalls (formula (3), p. 28) that

$$A(r,f)=\frac1\pi\iint_{\lvert z\rvert<r} \frac{\lvert f'(z)\rvert^2\,dx\,dy}{(1+\lvert f(z)\rvert^2)^2} =\frac1\pi\iint_{\overline{\mathbb C}}n(r,a,f)\,d\omega(a),$$

where $z=x+iy$ and $d\omega(a)$ is the element of area in the spherical metric.

Theorem (pp. 28--29, the paper's only theorem, unnumbered). There exists an entire function $f$ with the following three properties.

• (A) For all $a,b\in\mathbb C$ with $a\ne b$,

$$\varlimsup_{r\to\infty}\frac{n(r,a)}{n(r,b)}=\infty,\qquad \varliminf_{r\to\infty}\frac{n(r,a)}{n(r,b)}=0.\qquad(2)$$

• (B) For all $a\in\mathbb C$,

$$\varlimsup_{r\to\infty}\frac{n(r,a,f)}{A(r,f)}=\infty.\qquad(4)$$

• (C) There is a sequence $(r_k)$ with $r_k\to\infty$ such that, for all $a\in\mathbb C$,

$$\lim_{k\to\infty}\frac{n(r_k,a,f)}{A(r_k,f)}=0,\qquad(5)$$

and the convergence to zero is uniform with respect to the points $a$ of any bounded domain in $\mathbb C$.

The paper notes (p. 28) that since $a$ and $b$ in (2) are arbitrary, the second equality in (2) follows from the first. It also notes (p. 29), using (3), that for no sequence $r_k\to\infty$ can $n(r_k,a,f)/A(r_k,f)$ tend to $\infty$ for all $a$ in some set $D\subset\mathbb C$ of positive plane measure.

Consequence for the spherical deficiencies

For $f$ meromorphic in $\mathbb C$ the paper defines (p. 29)

$$\delta_S(a)=1-\varlimsup_{r\to\infty}\frac{n(r,a,f)}{A(r,f)},\qquad \Delta_S(a)=1-\varliminf_{r\to\infty}\frac{n(r,a,f)}{A(r,f)},$$

analogues of the Nevanlinna and Valiron deficiencies $\delta(a)$ and $\Delta(a)$, and records the inequalities $\delta_S(a)\le\delta(a)\le\Delta(a)\le\Delta_S(a)\le1$, from which Shimizu's defect relation $\sum_{a\in\overline{\mathbb C}}\delta_S^+(a)\le2$ follows. By (B) and (C), the function of the theorem has $\delta_S(a)=-\infty$ and $\Delta_S(a)=1$ for every $a\in\mathbb C$, so the analogy with $\delta(a)$ and $\Delta(a)$ does not extend far (p. 29).

Proof pointer

Pp. 29--35, Sections 1 to 7. Section 1 defines explicit domains $D(k,s,j)$ and $D_1(k,s,j)$ of the $w$-plane, orders their index triples into one sequence, and states a lemma: for distinct $a,b\in\mathbb C$ there is an increasing sequence of indices $n_\nu$ with $a$ in the $\nu$-th inner domain and $b$ outside the $\nu$-th domain and away from the boundaries of the domains of that step. Section 2 builds simply connected Riemann surfaces over a large disc by slits and glued branch surfaces, maps the unit disc onto them, records the number of $a$-points of each map over each region (formula (7)), and from these maps builds functions $F_j$ together with radii $r_j$. Section 3 shows that the power series coefficients of $F_j(z/\rho_j)$ converge to those of an entire function $f$ with $\lvert f(r_je^{i\varphi})-F_j(e^{i\varphi})\rvert$ small (formula (22)). Rouché's theorem then transfers the $a$-point counts to $f$, which gives (A) in Section 4; Ahlfors's first covering theorem, comparing $A(r,f)$ with the mean sheet number over a disc of radius 1 centred at $3$ or $-3$, gives (B) in Section 5. Section 6 gives (C) by a separate surface of the same kind, and Section 7 combines the two families of surfaces so that one entire function has (A), (B) and (C). The paper says its argument is close in several essential points to Hayman's method.

Read depth

Claims checked: the setting, the theorem and the consequence on pp. 28--29 were read clause by clause on the page images of the print. The proof was read for structure only. Nothing here is independently reviewed.

Source. A. A. Gol'dberg, Counting functions of sequences of $a$-points for entire functions (Russian), Sibirsk. Mat. Zh. 19 (1978), no. 1, 28--36, 236; the edition read is named on the source card.

Bears on

• Problem 1116: property (A) gives an entire function with $\varlimsup_{r\to\infty}n(r,a)/n(r,b)=\infty$ and $\varliminf_{r\to\infty}n(r,a)/n(r,b)=0$ for every pair of distinct finite values, which the paper (p. 28) calls an affirmative answer to Erdős's question, Problem 1.25 of Hayman's 1974 list of new problems. The paper says (p. 28) that the analogous question for functions meromorphic in $\mathbb C$, with $a,b$ in the extended plane, remains open; the theorem does not address it.