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import FormalConjecturesUtilErdős Problem 517
[Bi28] Biernacki, Miécislas, Sur les équations algébriques contenant des paramétres arbitraires. (1928), 145.
open Set Filter Topology
namespace Erdos517
If 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?
@[category research open, AMS 30]
theorem erdos_517 : answer(sorry) ↔ ∀ {f : ℂ → ℂ} {n : ℕ → ℕ} (hn : HasFabryGaps n)
{a : ℕ → ℂ} (ha : ∀ k, a k ≠ 0) (hf : ∀ z, HasSum (fun k => a k * z ^ n k) (f z)) (z : ℂ),
{x : ℂ | f x = z}.Infinite := ⊢ 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}.Infinite
All goals completed! 🐙
If f(z) = ∑ aₖzⁿₖ is an entire function (with aₖ ≠ 0 for all k) such that ∑ 1 / nₖ < ∞,
then f assumes every value infinitely often. This theorem is proved in [Bi28].
@[category research solved, AMS 30]
theorem erdos_517.variants.fejer {f : ℂ → ℂ} {n : ℕ → ℕ} (hn : HasFejerGaps n) {a : ℕ → ℂ}
(ha : ∀ k, a k ≠ 0) (hf : ∀ z, HasSum (fun k => a k * z ^ n k) (f z)) (z : ℂ) :
{x : ℂ | f x = z}.Infinite := f:ℂ → ℂn:ℕ → ℕhn:HasFejerGaps na:ℕ → ℂha:∀ (k : ℕ), a k ≠ 0hf:∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)z:ℂ⊢ {x | f x = z}.Infinite
All goals completed! 🐙
end Erdos517