519.lean`

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Project sources: Pinned project documentation, Formal Conjectures paper, arXiv v1

2 declarations, 1 contain sorry, 0 declared as axioms

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.

Imports (1)

FormalConjecturesUtil

def Erdos519.powerSum The $k$th power sum of a finite sequence of complex numbers line 46 source
Source documentation
The $k$th power sum of a finite sequence of complex numbers.

standard axioms: Classical.choice, Quot.sound, propext what this means
Statement uses: Nat, Fin, Complex
Proof uses: Finset.sum, Finset.univ, HPow.hPow
7 instances, coercions and auxiliary constants left out.
```
noncomputable def powerSum {n : ℕ} (z : Fin n → ℂ) (k : ℕ) : ℂ :=
…
```
theorem Erdos519.erdos_519 Let $z_1,\ldots,z_n\in \mathbb{C}$ with $z_1=1$ sorry line 59 source
Source documentation
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.

depends on sorry what this means
Statement uses: Iff, True, Exists, Real, And, LT.lt, OfNat.ofNat, Nat, Fin, Complex, Eq, Fin.mk, Norm.norm, Erdos519.powerSum, HAdd.hAdd
Proof uses: sorryAx, Lean.Name, Bool.false, Lean.Name.num, Lean.Name.str, Lean.Name.anonymous
22 instances, coercions and auxiliary constants left out.

Tactic trace: 1 steps
Open to load the checked before-and-after goal states.
```
theorem 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
…
```
Mentions: c, Erdos519.powerSum