Oriike 2026 negative answer universal function version erdos

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corollary_1: Oriike's corollary that no function Psi tending to infinity, chosen independently of f, has the property that every transcendental entire function f has a path to infinity along which |f| divided by Psi of the maximum modulus tends to infinity; the powers T^eps with eps > 0 fail in particular.

theorem_1: Oriike's theorem that for every nondecreasing function Phi tending to infinity there is a transcendental entire function f such that, along every path to infinity, the quotient of |f| by Phi of the maximum modulus has lower limit zero.

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Yuta Oriike, A Negative Answer to the Universal-Function Version of Erdős's Third Question in Problem 514. Unpublished note (revised draft, May 2026). No notice is printed in the file, and the source repository has no license file and states no license (https://github.com/yuta0x89/ErdosProblems, read 2026-10-02); the term is unstated.

The note addresses only the third question of Erdős Problem 514, taken in a universal reading: one comparison function, chosen before f and the same for every f. Theorem 1 states that for every nondecreasing Φ(T) → ∞ there is a transcendental entire function f such that every path to infinity γ has liminf |f(γ(t))|/Φ(M_f(|γ(t)|)) = 0, where M_f(r) is the maximum modulus; Corollary 1 concludes that no universal Ψ(T) → ∞ works, in particular no power Ψ(T) = T^ε. The note claims no priority for this power case, for which Chojecki used Langley's formulation of a consequence of Barth–Brannan–Hayman. The proof deduces Theorem 1 from Hayman's theorem on growth of entire functions along asymptotic paths (quoted as Theorem 2) once Hayman's auxiliary growth function is chosen suitably (Lemma 1), and a self-contained direct construction (Lemma 2, with sequences r_n, X_n, λ_n, a_n) gives a second, formalization-friendly proof. The note says that the first two questions of Problem 514 are closely related to the subharmonic path results of Huber, Lewis–Rossi–Weitsman and Wu, which Chojecki's note applies to them, and that the positive path-growth results of Rossi–Weitsman and Toda are stated in terms of |z| and tract geometry rather than as universal lower bounds in terms of M_f(r). For Problem 514 this note claims a negative answer to the universal-function version of the third question only; it does not reprove the first two questions.

Source: https://github.com/yuta0x89/ErdosProblems/blob/main/notes/514/erdos_514_third_question_forum_note_revised.pdf.

Bears on. #514, third question, read with the comparison function fixed before $f$: Theorem 1 (p. 2) gives, for each nondecreasing $\Phi(T)\to\infty$, a transcendental entire $f$ with no path on which $\lvert f\rvert/\Phi(M_f)$ tends to infinity, and Corollary 1 (p. 3) concludes that no fixed $\Psi(T)\to\infty$, including $\Psi(T)=T^\varepsilon$, works for every $f$. The note does not treat the first two questions.

Results to transcribe.

• Theorem 1 (p. 2): For every nondecreasing Φ with Φ(T) → ∞ there is a transcendental entire function f such that every path to infinity γ satisfies liminf_{t→∞} |f(γ(t))|/Φ(M_f(|γ(t)|)) = 0.
• Corollary 1 (p. 3): No universal comparison function Ψ(T) → ∞, independent of f, admits for every transcendental entire function a path with |f(z)|/Ψ(M_f(|z|)) → ∞; the power case Ψ(T) = T^ε fails in particular.
• Theorem 2 (Hayman, p. 3): Quoted growth theorem for entire functions along asymptotic paths in terms of a positive increasing auxiliary function λ(r), used as the main input.
• Lemma 1 (p. 3): For a nondecreasing Φ with Φ(T) → ∞ there is a positive increasing λ(r) with log λ(r)/log r → ∞ such that, for every positive integer k, Φ(exp(exp(λ(r)))) ≥ exp(r^k) for all sufficiently large r; this reduces Theorem 1 to Hayman's theorem.
• Lemma 2 (p. 5): The parameters r_n, X_n, λ_n, a_n of the direct construction can be chosen to satisfy the off-diagonal and size conditions (1), (2) and (3), giving a self-contained proof of Theorem 1.

No file of this source is held: no license on record permits its redistribution, and the card cites the edition it names above.