FormalConjectures.ErdosProblems.«519»

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import FormalConjecturesUtil

Erdős Problem 519

References:

erdosproblems.com/519

[Er61] Erdős, Paul, Some unsolved problems. Magyar Tud. Akad. Mat. Kutató Int. Közl. (1961), 221-254.

[Er65b] Erdős, Paul, Some recent advances and current problems in number theory. Lectures on Modern Mathematics, Vol. III (1965), 196-244.

[Bi94] Biró, A., On a problem of Turán concerning sums of powers of complex numbers. Acta Math. Hungar. (1994), 209-216.

[Bi00] Biró, András, An improved estimate in a power sum problem of Turán. Indag. Math. (N.S.) (2000), 343-358.

[Bi00b] Biró, A., An upper estimate in Turán's pure power sum problem. Indag. Math. (N.S.) (2000), 499--508.

[At61b] Atkinson, F. V., On sums of powers of complex numbers. Acta Math. Acad. Sci. Hungar. (1961), 185-188.


open scoped BigOperators


namespace Erdos519

The $k$th power sum of a finite sequence of complex numbers.

noncomputable def powerSum {n : ℕ} (z : Fin n → ℂ) (k : ℕ) : ℂ :=
  ∑ i : Fin n, z i ^ k

Let $z_1,\ldots,z_n\in \mathbb{C}$ with $z_1=1$. Must there exist an absolute constant $c>0$ such that $$ \max_{1\leq k\leq n}\left\lvert \sum_{i}z_i^k\right\rvert>c? $$

Atkinson proved that $c=1/6$ suffices.

@[category research solved, AMS 30, formal_proof using lean4 at "https://github.com/plby/lean-proofs/blob/main/src/v4.29.1/ErdosProblems/Erdos519.lean"]
theorem declaration uses 'sorry'erdos_519 : answer(True) ↔
    ∃ c : ℝ, 0 < c ∧
      ∀ (n : ℕ) (hn : 0 < n) (z : Fin n → ℂ),
        z ⟨0, hn⟩ = 1 →
          ∃ k : Fin n, c < ‖powerSum z (k.val + 1)‖ := by⊢ True ↔ ∃ c, 0 < c ∧ ∀ (n : ℕ) (hn : 0 < n) (z : Fin n → ℂ), z ⟨0, hn⟩ = 1 → ∃ k, c < ‖powerSum z (↑k + 1)‖
  sorryAll goals completed! 🐙


end Erdos519