Problem 514

Let $f(z)$ be an entire transcendental function. Does there exist a path $L$ so that, for every $n$ ,

\lvert f(z)/z^n\rvert \to \infty

as $z\to \infty$ along $L$ ?

Can the length of this path be estimated in terms of $M(r)=\max_{\lvert z\rvert=r}\lvert f(z)\rvert$ ? Does there exist a path along which $\lvert f(z)\rvert$ tends to $\infty$ faster than a fixed function of $M(r)$ (such that $M(r)^\epsilon$ )?

Source: erdosproblems.com/514

Analysis

Wiki page

Status

ClaimedAnswered

A full solution has been claimed but not yet accepted. Settled in another form, for example when its parts resolve differently or the question is open-ended.

The site labels the problem OPEN (page last edited 18 January 2026) and credits Boas, unpublished, with the first question. The three parts are answered by three partial claims: the first question yes, by the accepted claim Lewis, Rossi and Weitsman 1984, on its refereed publication; the first and second yes, by the pending claim Chojecki 2026, a note of 20 April 2026 produced with GPT-5.4 Pro; and the third no, by the pending claim Oriike 2026, a note of 28 April 2026 produced with GPT-5.5 Pro, with a Lean file. Accepted and pending claims together settle every part, so the derived standing is claimed, and its value is answered because the parts are answered in opposite directions. The site has adopted none of the claims.

Ch26: Chojecki, P., A note on an Erdős path problem for transcendental entire functions. Note (2026), https://www.ulam.ai/research/erdos514.pdf.
Er61: Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.
Er80: Eremenko, A. È., Growth of entire and subharmonic functions on asymptotic curves. Sib. Math. J. 21 (1980), no. 5, 673-683.
GoEr80: Gol'dberg, A. A. and Eremenko, A. È., On asymptotic curves of entire functions of finite order. Math. USSR-Sb. 37 (1980), 509-533.
Ha60: Hayman, W. K., On the growth of integral functions on asymptotic paths. J. Indian Math. Soc. (N.S.) 24 (1960), 251-264.
La22: Langley, J. K., Complex flows, escape to infinity and a question of Rubel. Ann. Fenn. Math. 47 (2022), 885-894.
LRW84: Lewis, John and Rossi, John and Weitsman, Allen, On the growth of subharmonic functions along paths. Ark. Mat. (1984), 109-119.
Or26: Oriike, Yuta, A negative answer to the universal-function version of Erdős's third question in Problem 514. Note (2026), revised May 2026.
Ta76: Talpur, M. N. M., On the growth of subharmonic functions on asymptotic paths. Proc. London Math. Soc. (3) 32 (1976), no. 2, 193-198.
Wu85: Wu, Jang-Mei, Length of paths for subharmonic functions. J. London Math. Soc. (2) 32 (1985), 497-505.

Problem 514

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