FormalConjectures.ErdosProblems.«516»

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import FormalConjecturesUtil

Erdős Problem 516

References:

erdosproblems.com/516

[Fu63] Fuchs, W. H. J., Proof of a conjecture of G. Pólya concerning gap series. Illinois J. Math. (1963), 661--667.

[Ko65] Kövari, Thomas, A gap-theorem for entire functions of infinite order. Michigan Math. J. (1965), 133--140.


open scoped Nat

open Filter Real Set


namespace Erdos516

An entire function f is said to be of finite order if there exist numbers c, a ≥ 0 such that for all z, ‖f z‖ ≤ c * rexp (‖z‖ ^ a).

def OfFiniteOrder {E F: Type*} [NormedAddCommGroup E] [NormedSpace ℂ E]
    [NormedAddCommGroup F] [NormedSpace ℂ F] (f : E → F) : Prop :=
  Differentiable ℂ f ∧ ∃ c ≥ 0, ∃ a ≥ 0, ∀ z, ‖f z‖ ≤ c * rexp (‖z‖ ^ a)


noncomputable def ratio (r : ℝ) (f : ℂ → ℂ) : ℝ :=
  (⨅ z : {z : ℂ // ‖z‖ = r}, ‖f z‖).log / (⨆ z : {z : ℂ // ‖z‖ = r}, ‖f z‖).log

Let f = ∑ aₖzⁿₖ be an entire function of finite order such that nₖ / k → ∞. Then limsup (fun r => ratio r f) atTop = 1. This is proved in [Fu63].

@[category research solved, AMS 30]
theorem declaration uses 'sorry'erdos_516 {f : ℂ → ℂ} {n : ℕ → ℕ}
    (hn : HasFabryGaps n) {a : ℕ → ℂ} (ha : ∀ n, a n ≠ 0)
    (hfn : ∀ z, HasSum (fun k => a k * z ^ n k) (f z)) (hf : OfFiniteOrder f) :
    limsup (fun r => ratio r f) atTop = 1 := byf:ℂ → ℂn:ℕ → ℕhn:HasFabryGaps na:ℕ → ℂha:∀ (n : ℕ), a n ≠ 0hfn:∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)hf:OfFiniteOrder f⊢ limsup (fun r => ratio r f) atTop = 1
  sorryAll goals completed! 🐙

Let f = ∑ aₖzⁿₖ be an entire function such that nₖ > k (log k) ^ (2 + c). Then limsup (fun r => ratio r f) atTop = 1. This is proved in [Ko65].

@[category research solved, AMS 30]
theorem declaration uses 'sorry'erdos_516.variants.limsup_ratio_eq_one {f : ℂ → ℂ} {n : ℕ → ℕ}
    (hn : ∃ c > (0 : ℝ), ∀ k, n k > k * log k ^ (2 + c)) {a : ℕ → ℂ} (ha : ∀ n, a n ≠ 0)
    (hfn : ∀ z, HasSum (fun k => a k * z ^ n k) (f z)) :
    limsup (fun r => ratio r f) atTop = 1 := byf:ℂ → ℂn:ℕ → ℕhn:∃ c > 0, ∀ (k : ℕ), ↑(n k) > ↑k * log ↑k ^ (2 + c)a:ℕ → ℂha:∀ (n : ℕ), a n ≠ 0hfn:∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)⊢ limsup (fun r => ratio r f) atTop = 1
  sorryAll goals completed! 🐙

Is it true that for all entire functions f = ∑ aₖzⁿₖ such that ∑' 1 / nₖ < ∞, limsup (fun r => ratio r f) atTop = 1?

@[category research open, AMS 30]
theorem declaration uses 'sorry'erdos_516.variants.limsup_ratio_eq_one_of_hasFejerGaps : answer(sorry) ↔
    ∀ {f : ℂ → ℂ} {n : ℕ → ℕ} (hn : HasFejerGaps n) {a : ℕ → ℂ} (ha : ∀ n, a n ≠ 0)
    (hfn : ∀ z, HasSum (fun k => a k * z ^ n k) (f z)),
    limsup (fun r => ratio r f) atTop = 1 := by⊢ True ↔
  ∀ {f : ℂ → ℂ} {n : ℕ → ℕ},
    HasFejerGaps n →
      ∀ {a : ℕ → ℂ},
        (∀ (n : ℕ), a n ≠ 0) →
          (∀ (z : ℂ), HasSum (fun k => a k * z ^ n k) (f z)) → limsup (fun r => ratio r f) atTop = 1
  sorryAll goals completed! 🐙


end Erdos516