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Claim. The answer is yes for an entire function and finite values. The Theorem of the paper (p. 28) constructs an entire $f$ with three properties: (A) for all $a,b\in\mathbb C$ with $a\ne b$, $\limsup_{r\to\infty}n(r,a)/n(r,b)=\infty$ and $\liminf_{r\to\infty}n(r,a)/n(r,b)=0$, the second following from the first because the pair is arbitrary; (B) $\limsup_{r\to\infty}n(r,a,f)/A(r,f)=\infty$ for every $a\in\mathbb C$, where $A(r,f)$ is the mean number of sheets of the Riemann surface onto which $f$ maps $|z|<r$; (C) a sequence $r_k\to\infty$ with $n(r_k,a,f)/A(r_k,f)\to0$ for every $a$, uniformly on bounded sets. Property (A) is the problem's property. The introduction presents it as the affirmative answer to Erdős's question, Problem 1.25 of Hayman's collection, and adds that for meromorphic functions with $a,b$ in the extended plane the analogous question remained open; that case is settled by Toppila's Theorem 2 (Toppila's page). The construction uses explicit domains in the $w$-plane and follows Hayman's method closely. The statement is as printed in the paper, whose source card is goldberg_1978_counting_functions_sequences_points_entire_functions; the proof is not verified in this corpus.

Covers. The entire-function reading of the question, for finite values $a\ne b$: an entire $f$ has $n(r,\infty)=0$, so it cannot satisfy the meromorphic reading, in which $a$ or $b$ may be $\infty$. That reading is settled by Toppila's Theorem 2 on Toppila's page, which carries the problem's standing.

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 publisher's record is the linked mathnet.ru page. The record dates the article to the year alone, and the page is named by that date.

Acceptance. Refereed: the paper appeared in Sibirskii Matematicheskii Zhurnal. Reviewed: the site's curator, T. F. Bloom, labels the problem solved and credits Gol'dberg and Toppila with constructing entire functions with the property; Hayman and Lingham's 2018 survey of Hayman's problems (library card, Update 1.25) records both entire examples. Nothing here is independently reviewed by this project.

Depends on. Nothing on the wiki.