Erdős Problem #513

Partial progress by GPT-5.2 Thinking (1 Mar 2026): slight improvement to past construction.

Problem
Erdős problem #513 (analysis); current status: open
Credit
GPT-5.2 Thinking · AI building on known literature
Checked
Listed in Tao's wiki
Timeline
First AI contribution 2026-03-01 · wiki data as of 2026-06-30
Sources
Problem page (erdosproblems.com)Tao's AI-contributions wiki
PARTIALPer Tao's AI-contributions wiki

The problem

Let f=∑n=0∞anznf=\sum_{n=0}^\infty a_nz^n be a transcendental entire function. What is the greatest possible value of

lim inf⁡r→∞max⁡n∣anrn∣max⁡∣z∣=r∣f(z)∣?\liminf_{r\to \infty} \frac{\max_n\lvert a_nr^n\rvert}{\max_{\lvert z\rvert=r}\lvert f(z)\rvert}?

Statement quoted from erdosproblems.com/513, maintained by Thomas Bloom. Original source: [Er61,p.249]. Current status there: open.

AI contributions

As recorded in Terence Tao's wiki AI contributions to Erdős problems (data as of 30 Jun 2026). 🟢 full, 🟡 partial, ⚪ unverified, 🔴 incorrect.

Date Kind AI systems Outcome Literature
1 Mar, 2026 AI building on known literature GPT-5.2 Thinking 🟡 Slight improvement to past construction 🟡 He and Tang (2026)

Caveats

The wiki's classifications are provisional and are not a benchmark. Many Erdős problems have never had a thorough literature review, so a result that looks new may later turn out to be known; some problems were also stated ambiguously or incorrectly. See the problem page for the full discussion.

∎