| import OAI.NumberTheory.Jacobsthal.Primes.UniformDirichletBasepoint |
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| namespace OAI |
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| namespace Erdos970 |
| open scoped _root_.Erdos970 |
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| section |
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| namespace Erdos970Dependency.SiegelWalfisz |
| open _root_.Filter |
| open scoped Topology BigOperators |
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| theorem exists_local_zero_log_derivative_constant : |
| ∃ C : ℝ, 0 < C ∧ ∀ (B : ℝ), 1 < B → ∀ f : ℂ → ℂ, |
| (∀ z ∈ Metric.closedBall (0 : ℂ) 1, AnalyticAt ℂ f z) → |
| f 0 = 1 → (∀ z ∈ Metric.closedBall (0 : ℂ) 1, ‖f z‖ ≤ B) → |
| ∃ S : Finset ℂ, (∀ z, z ∈ S ↔ ‖z‖ ≤ 19/20 ∧ f z = 0) ∧ |
| ∀ z : ℂ, ‖z‖ ≤ 17/20 → f z ≠ 0 → |
| ‖deriv f z / f z - ∑ rho ∈ S, (analyticOrderAt f rho).toNat / (z-rho)‖ ≤ |
| C * Real.log B := by |
| classical |
| let K : ℝ := 16 * (9/10)^2 / (((9/10)-(17/20))^3) + |
| 1 / (((39/40)^2/(19/20)-(19/20)) * Real.log ((39/40)/(19/20))) |
| refine ⟨max 1 K, lt_of_lt_of_le zero_lt_one (le_max_left _ _), ?_⟩ |
| intro B hB f hf hf0 hfB |
| have hfne : f 0 ≠ 0 := by rw [hf0]; norm_num |
| have hfinite : (_root_.Erdos970.zerosetKfR (19/20) (by norm_num) f).Finite := |
| _root_.Erdos970.lem_Contra_finiteKR (19/20) (by norm_num) (by norm_num) f hf ⟨0, by simp, hfne⟩ |
| have hex (rho : ℂ) : ∃ g : ℂ → ℂ, |
| rho ∈ _root_.Erdos970.zerosetKfR (19/20) (by norm_num) f → |
| AnalyticAt ℂ g rho ∧ g rho ≠ 0 ∧ |
| ∀ᶠ z in 𝓝 rho, f z = (z-rho)^(analyticOrderAt f rho).toNat * g z := by |
| by_cases hr : rho ∈ _root_.Erdos970.zerosetKfR (19/20) (by norm_num) f |
| · obtain ⟨g, hg⟩ := _root_.Erdos970.lem_analytic_zero_factor (39/40) (19/20) |
| (by norm_num) (by norm_num) (by norm_num) f hf hfne rho hr |
| exact ⟨g, fun _ => hg⟩ |
| · exact ⟨fun _ => 1, fun h => (hr h).elim⟩ |
| choose g hg using hex |
| refine ⟨hfinite.toFinset, ?_, ?_⟩ |
| · intro z |
| simp only [Set.Finite.mem_toFinset, _root_.Erdos970.zerosetKfR, Set.mem_ofPred_eq, |
| Metric.mem_closedBall, dist_zero_right] |
| · intro z hz hzne |
| have hz' : z ∈ Metric.closedBall (0 : ℂ) (17/20) \ |
| _root_.Erdos970.zerosetKfR (19/20) (by norm_num) f := by |
| refine ⟨by simpa only [Metric.mem_closedBall, dist_zero_right] using hz, ?_⟩ |
| intro h |
| exact hzne h.2 |
| have h := _root_.Erdos970.final_ineq1 B hB (17/20) (9/10) (39/40) (19/20) |
| (by norm_num) (by norm_num) (by norm_num) (by norm_num) (by norm_num) |
| f hf hf0 hfinite hg |
| (fun w hw => hfB w ((Metric.closedBall_subset_closedBall (by norm_num : (39/40:ℝ) ≤ 1)) hw)) z hz' |
| apply h.trans |
| exact mul_le_mul_of_nonneg_right (le_max_right (1:ℝ) K) (Real.log_pos hB).le |
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| end Erdos970Dependency.SiegelWalfisz |
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| end |
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| end Erdos970 |
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| end OAI |
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