| import OAI.NumberTheory.Jacobsthal.Estimates.SourceCenteredGap |
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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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| open _root_.Set _root_.Filter |
| open scoped Topology |
| namespace ErdosSourceReferenceMargin |
| open NumberTheoryLean.ReferenceMertens NumberTheoryLean.ReferenceProductErrors |
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| theorem root_B_log_W {z : ℝ} (hw : 1 < ErdosInverseBoxHeight.sourceW z) : |
| ErdosInverseBoxHeight.sourceB z*Real.log (ErdosInverseBoxHeight.sourceW z)=Real.log z := by |
| exact div_mul_cancel₀ _ (Real.log_pos hw).ne' |
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| theorem root_product_error_decay {c : ℝ} (hc : 0 < c) : |
| Tendsto (fun z : ℝ => Real.exp (-c*Real.sqrt (Real.log z))) atTop (𝓝 0) := by |
| have ht := (Real.tendsto_sqrt_atTop.comp Real.tendsto_log_atTop).const_mul_atTop hc |
| simpa only [Function.comp_def,neg_mul] using! Real.tendsto_exp_neg_atTop_nhds_zero.comp ht |
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| theorem root_referenceProduct_error_tendsto (closed : Bool) : |
| Tendsto (fun z => ErdosInverseBoxHeight.sourceB z* |
| referenceProduct (ErdosInverseBoxHeight.sourceW z) (ErdosInverseBoxHeight.sourceB z) closed- |
| normalization (ErdosInverseBoxHeight.sourceW z)) atTop (𝓝 0) := by |
| obtain ⟨c,C,w₀,hc,hC,hw₀,h⟩ := reference_product_absolute |
| have hlim : Tendsto (fun z : ℝ => C*Real.exp (-c*Real.sqrt (Real.log z))) atTop (𝓝 0) := by |
| simpa only [mul_zero] using (root_product_error_decay hc).const_mul C |
| have hbound : ∀ᶠ z : ℝ in atTop, |
| |ErdosInverseBoxHeight.sourceB z*referenceProduct (ErdosInverseBoxHeight.sourceW z) |
| (ErdosInverseBoxHeight.sourceB z) closed-normalization (ErdosInverseBoxHeight.sourceW z)| ≤ |
| C*Real.exp (-c*Real.sqrt (Real.log z)) := by |
| filter_upwards [rootB_tendsto.eventually (eventually_ge_atTop (2:ℝ)), |
| rootW_tendsto.eventually (eventually_ge_atTop w₀)] with z hb hw |
| have hW : 1 < ErdosInverseBoxHeight.sourceW z := hw₀.trans_le hw |
| have hB : 0 < ErdosInverseBoxHeight.sourceB z := by linarith |
| have hp := (h (ErdosInverseBoxHeight.sourceW z) hw).2.2 (ErdosInverseBoxHeight.sourceB z) hb closed |
| rw [root_B_log_W hW] at hp |
| have hh := mul_le_mul_of_nonneg_left hp hB.le |
| have he : ErdosInverseBoxHeight.sourceB z*referenceProduct (ErdosInverseBoxHeight.sourceW z) |
| (ErdosInverseBoxHeight.sourceB z) closed-normalization (ErdosInverseBoxHeight.sourceW z)= |
| ErdosInverseBoxHeight.sourceB z*(referenceProduct (ErdosInverseBoxHeight.sourceW z) |
| (ErdosInverseBoxHeight.sourceB z) closed-normalization (ErdosInverseBoxHeight.sourceW z)/ErdosInverseBoxHeight.sourceB z) := by |
| field_simp |
| rw [he,abs_mul,abs_of_pos hB] |
| have hcanc : ErdosInverseBoxHeight.sourceB z* |
| ((C/ErdosInverseBoxHeight.sourceB z)*Real.exp (-c*Real.sqrt (Real.log z)))= |
| C*Real.exp (-c*Real.sqrt (Real.log z)) := by field_simp |
| exact hh.trans_eq hcanc |
| have habs := squeeze_zero' (Filter.Eventually.of_forall (fun z => abs_nonneg |
| (ErdosInverseBoxHeight.sourceB z*referenceProduct (ErdosInverseBoxHeight.sourceW z) |
| (ErdosInverseBoxHeight.sourceB z) closed-normalization (ErdosInverseBoxHeight.sourceW z)))) hbound hlim |
| apply tendsto_zero_iff_norm_tendsto_zero.mpr |
| simpa only [Real.norm_eq_abs] using habs |
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| theorem root_referenceProduct_tendsto (closed : Bool) : |
| Tendsto (fun z => ErdosInverseBoxHeight.sourceB z* |
| referenceProduct (ErdosInverseBoxHeight.sourceW z) (ErdosInverseBoxHeight.sourceB z) closed) atTop (𝓝 1) := by |
| have hh := (root_referenceProduct_error_tendsto closed).add (normalization_tendsto_one.comp rootW_tendsto) |
| simpa only [sub_add_cancel,zero_add,Function.comp_def] using! hh |
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| end ErdosSourceReferenceMargin |
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| end |
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| end Erdos970 |
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| end OAI |
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