Complex functions
f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that nₖ / k → ∞,
is it true that f assumes every value infinitely often? $4,969 bountynobody has started
Nobody has attempted this. A complete proof or refutation takes the whole bounty.
Even one lemma helps. Whoever finishes this later shares the pool with everyone who got them there.
contrib new erdos-517Command line, then a pull request on GitHub.
Nothing has been published against this problem yet.
A first lemma is worth as much as a last one: whoever closes the problem shares the pool with everyone who got them there.
Formal statement
Proof target
import FormalConjectures.ErdosProblems.«517»
import TaskSupport
namespace Bounty
theorem target : fcTypeOfName% "Erdos517.erdos_517" := by
sorry
end Bounty
Original conjecture (Lean type)
True ↔
∀ {f : ℂ → ℂ} {n : ℕ → ℕ},
HasFabryGaps n →
∀ {a : ℕ → ℂ},
(∀ (k : ℕ), a k ≠ 0) → (∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)) → ∀ (z : ℂ), {x | f x = z}.InfiniteOriginal conjecture source: FormalConjectures/ErdosProblems/517.lean
References
Something wrong with this formalization?
A statement that does not faithfully capture the original conjecture is the one real risk here, so we would rather hear about it early - before someone spends weeks on it.
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