Goldberg 1978 counting functions sequences points entire functions

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theorem: Gol'dberg's theorem that some entire function f has, for all distinct complex a and b, upper limit infinity and lower limit zero of n(r,a)/n(r,b), together with an upper limit infinity of n(r,a,f)/A(r,f) for every complex a and a sequence r_k along which that ratio tends to zero for every complex a, uniformly on bounded domains.

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Gol'dberg, A. A., Counting functions of sequences of a-points for entire functions (Russian). Sibirsk. Mat. Zh. 19 (1978), no. 1, 28--36, 236.

In Russian. Nevanlinna theory gives N(r,a) ~ T(r,f) for all a outside a small exceptional set, so lim N(r,a)/N(r,b) = 1 for all a, b outside that set. Gol'dberg shows that the analog fails completely for the unintegrated counting functions n(r,a): he constructs an entire f such that for all a, b in C with a not equal b, one has limsup n(r,a)/n(r,b) = infinity and liminf n(r,a)/n(r,b) = 0 (formula (2)), which gives an affirmative answer to a question of P. Erdos (cited as Problem 1.25 of a problem collection). The single main Theorem produces an entire f with three properties: (A) property (2) holds for all distinct a, b in C; (B) for all a in C, limsup n(r,a,f)/A(r,f) = infinity, where A(r,f) is the mean number of sheets of the Riemann surface over which the disc |z| < r is mapped; and (C) there is a sequence r_k tending to infinity with lim n(r_k,a,f)/A(r_k,f) = 0 for all a in C, uniformly for a in any bounded domain. The construction uses a family of explicit domains D(k,s,j) and D_1(k,s,j) in the finite w-plane (the s = 2 domains are discs about the origin), ordered along a sequence of triples, with the argument close in several essential points to Hayman's method. Consequences are drawn for the analogs delta_S(a) and Delta_S(a) of the Nevanlinna and Valiron deficiencies: the defect relation sum of delta_S^+(a) <= 2 holds (Shimizu; it follows from delta_S(a) <= delta(a) <= Delta(a) <= Delta_S(a) <= 1), but (B) and (C) show the analogy with the Nevanlinna and Valiron deficiencies does not extend far, through an entire f with delta_S(a) = -infinity and Delta_S(a) = 1 for every finite a in C. The paper answers Problem 1116 (Erdos's question, problem 1.25 of Hayman's 1974 collection) for entire f and finite a not equal b; p. 28 leaves the meromorphic case with a, b in the extended plane open.

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Bears on. #1116: property (A) of the theorem (pp. 28--29) 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 all distinct $a,b\in\mathbb C$, which the paper calls an affirmative answer to Erdős's question (Problem 1.25 of Hayman's 1974 list) for entire functions and finite values; the paper leaves the meromorphic case, with $a,b$ in the extended plane, open (p. 28).

Results.

• Theorem (pp. 28--29, unnumbered): there exists an entire $f$ with (A) $\varlimsup n(r,a)/n(r,b)=\infty$ and $\varliminf n(r,a)/n(r,b)=0$ for all distinct $a,b\in\mathbb C$; (B) $\varlimsup n(r,a,f)/A(r,f)=\infty$ for all $a\in\mathbb C$; (C) a sequence $r_k\to\infty$ with $n(r_k,a,f)/A(r_k,f)\to0$ for all $a\in\mathbb C$, uniformly on bounded domains. The same page records the consequence (p. 29) that this $f$ has $\delta_S(a)=-\infty$ and $\Delta_S(a)=1$ for every $a\in\mathbb C$, while Shimizu's defect relation $\sum\delta_S^+(a)\le2$ still holds.

Read status: claims checked for the setting, the theorem and the consequence on pp. 28--29, read clause by clause on the page images; the proof (pp. 29--35) read for structure only. Nothing here is independently reviewed.

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