Erdős Problem #513
OPENdatabase updated 2025-08-31
No proof or disproof is known, and it isn't known whether a finite computation could settle it.
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The **Erdős maximum term constant** \(B\): the supremum, over all transcendental entire functions \(f\), of \(\liminf_{r \to \infty} \mu(r, f) / M(r, f)\). Here \(\mu(r, f)\) is the maximum term of the power series of \(f\) on \(|z| = r\), and \(M(r, f) = \max_{|z| = r} |f(z)|\).
Let `f` be a transcendental entire function. What is the greatest possible value of `liminf (fun r : ℝ => ratio r f) atTop`?
Statement as formalized in Formal Conjectures (Google DeepMind, Apache-2.0) · their label: research open · 2 further variants / known partial results in the Lean file. The authoritative wording is on erdosproblems.com/513.Prizenone
Topicsanalysis
Statement in LeanFormal Conjectures
Solution in Leanno
Original pageerdosproblems.com/513
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An explicit He–Tang-family entire function and independent Arb checker rigorously improve the known lower bound to B > 0.585078819674; the exact value remains open.
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