| import Mathlib.MeasureTheory.Integral.IntegralEqImproper |
| import OAI.NumberTheory.Jacobsthal.Estimates.PairedDeficitPositivity |
| import OAI.NumberTheory.Jacobsthal.Sieve.BuchstabBridge |
|
|
| namespace OAI |
|
|
| namespace Erdos970 |
|
|
| section |
|
|
| namespace NumberTheoryLean.NormalizedDeficits |
|
|
| open Filter Set |
| open scoped Topology |
| open LinearSieveFunctions BuchstabBridge |
|
|
| noncomputable def lowerDeficit (s : ℝ) : ℝ := 1 - f sieveA s |
| noncomputable def upperDeficit (s : ℝ) : ℝ := F sieveA s - 1 |
| noncomputable def gap (s : ℝ) : ℝ := F sieveA s - f sieveA s |
|
|
| theorem deficit_sum (s : ℝ) : lowerDeficit s + upperDeficit s = gap s := by |
| unfold lowerDeficit upperDeficit gap |
| ring |
|
|
| theorem gap_eq_rho {s : ℝ} (hs : 1 ≤ s) : gap s = sieveA * Dickman.rho (s - 1) / s := |
| gap_identity sieveA hs |
|
|
| theorem lowerDeficit_continuous : Continuous lowerDeficit := |
| continuous_const.sub (f_continuous sieveA) |
|
|
| theorem upperDeficit_continuousOn : ContinuousOn upperDeficit (Ici 1) := |
| ((F_continuousOn sieveA).mono (by intro s hs; change 0 < s; exact lt_of_lt_of_le zero_lt_one hs)).sub continuousOn_const |
|
|
| theorem gap_continuousOn : ContinuousOn gap (Ici 1) := by |
| have h := lowerDeficit_continuous.continuousOn.add upperDeficit_continuousOn |
| have heq : gap = fun s => lowerDeficit s + upperDeficit s := funext (fun s => (deficit_sum s).symm) |
| rw [heq] |
| exact h |
|
|
| theorem gap_pos {s : ℝ} (hs : 1 ≤ s) : 0 < gap s := |
| LinearSieveFunctions.gap_pos sieveA_pos hs |
|
|
| theorem lowerDeficit_mass_deriv {s : ℝ} (hs : 2 < s) : |
| HasDerivAt (fun t => t * lowerDeficit t) (-upperDeficit (s - 1)) s := by |
| convert! (hasDerivAt_id s).sub (mass_f_hasDerivAt sieveA hs) using 1 |
| · ext t |
| change t * (1 - f sieveA t) = t - t * f sieveA t |
| ring |
| · simp only [upperDeficit] |
| ring |
|
|
| theorem upperDeficit_mass_deriv {s : ℝ} (hs : 2 < s) : |
| HasDerivAt (fun t => t * upperDeficit t) (-lowerDeficit (s - 1)) s := by |
| convert! (mass_F_hasDerivAt sieveA hs).sub (hasDerivAt_id s) using 1 |
| · ext t |
| change t * (F sieveA t - 1) = t * F sieveA t - t |
| ring |
| · simp only [lowerDeficit] |
| ring |
|
|
| theorem gap_mass_deriv {s : ℝ} (hs : 2 < s) : |
| HasDerivAt (fun t => t * gap t) (-gap (s - 1)) s := by |
| convert! (mass_F_hasDerivAt sieveA hs).sub (mass_f_hasDerivAt sieveA hs) using 1 |
| · ext t |
| change t * (F sieveA t - f sieveA t) = t * F sieveA t - t * f sieveA t |
| ring |
| · simp only [gap] |
| ring |
|
|
| theorem lowerDeficit_tendsto_zero : Tendsto lowerDeficit atTop (𝓝 0) := by |
| convert! normalized_f_tendsto_one.const_sub 1 using 1 |
| simp |
|
|
| theorem upperDeficit_tendsto_zero : Tendsto upperDeficit atTop (𝓝 0) := by |
| convert! normalized_F_tendsto_one.sub_const 1 using 1 |
| simp |
|
|
| theorem gap_tendsto_zero : Tendsto gap atTop (𝓝 0) := pair_gap_tendsto_zero sieveA |
|
|
| theorem deficits_pos {s : ℝ} (hs : 1 ≤ s) : 0 < lowerDeficit s ∧ 0 < upperDeficit s := by |
| apply PairedDeficitPositivity.paired_pos lowerDeficit_continuous.continuousOn |
| upperDeficit_continuousOn ?_ ?_ ?_ (fun _ h => lowerDeficit_mass_deriv h) |
| (fun _ h => upperDeficit_mass_deriv h) lowerDeficit_tendsto_zero upperDeficit_tendsto_zero s hs |
| · intro t ht |
| simp only [lowerDeficit, f_initial sieveA ht.2, sub_zero, zero_lt_one] |
| · intro t ht |
| have ht0 : 0 < t := lt_of_lt_of_le zero_lt_one ht.1 |
| change 0 < F sieveA t - 1 |
| rw [F_initial sieveA ht0 (by linarith [ht.2])] |
| have h := (lt_div_iff₀ ht0).mpr (show 1 * t < sieveA by linarith [sieveA_gt_two, ht.2]) |
| linarith |
| · intro t ht |
| rw [deficit_sum] |
| exact gap_pos ht |
|
|
| theorem deficit_offsets_pos {δ : ℝ} (hhalf : δ < 1 / 2) |
| (hδA : δ * sieveA < 1) (hδA2 : δ * sieveA < sieveA - 2) |
| {s : ℝ} (hs : 1 ≤ s) : |
| 0 < lowerDeficit s - δ * gap s ∧ 0 < upperDeficit s - δ * gap s := by |
| let a : ℝ → ℝ := fun t => lowerDeficit t - δ * gap t |
| let c : ℝ → ℝ := fun t => upperDeficit t - δ * gap t |
| have ha : ContinuousOn a (Ici 1) := |
| lowerDeficit_continuous.continuousOn.sub (continuousOn_const.mul gap_continuousOn) |
| have hc : ContinuousOn c (Ici 1) := |
| upperDeficit_continuousOn.sub (continuousOn_const.mul gap_continuousOn) |
| have hginit : ∀ t ∈ Icc 1 2, gap t = sieveA / t := by |
| intro t ht |
| simp only [gap, f_initial sieveA ht.2, sub_zero, |
| F_initial sieveA (s := t) (by linarith [ht.1]) (by linarith [ht.2])] |
| have hai : ∀ t ∈ Icc 1 2, 0 < a t := by |
| intro t ht |
| have ht0 : 0 < t := lt_of_lt_of_le zero_lt_one ht.1 |
| have hdiv : δ * sieveA / t < 1 := (div_lt_one ht0).mpr (lt_of_lt_of_le hδA ht.1) |
| dsimp [a, lowerDeficit] |
| rw [f_initial sieveA ht.2, hginit t ht] |
| have heq : δ * (sieveA / t) = δ * sieveA / t := by ring |
| rw [heq] |
| linarith |
| have hci : ∀ t ∈ Icc 1 2, 0 < c t := by |
| intro t ht |
| have ht0 : 0 < t := lt_of_lt_of_le zero_lt_one ht.1 |
| have hdiv : 0 < (sieveA - δ * sieveA - t) / t := |
| div_pos (by linarith [ht.2]) ht0 |
| dsimp [c, upperDeficit] |
| rw [F_initial sieveA ht0 (by linarith [ht.2]), hginit t ht] |
| convert! hdiv using 1 |
| field_simp |
| ring |
| have hsum : ∀ t, 1 ≤ t → 0 < a t + c t := by |
| intro t ht |
| have heq : a t + c t = (1 - 2 * δ) * gap t := by |
| dsimp [a, c] |
| rw [← deficit_sum t] |
| ring |
| rw [heq] |
| exact mul_pos (by linarith) (gap_pos ht) |
| have hda : ∀ t, 2 < t → HasDerivAt (fun u => u * a u) (-c (t - 1)) t := by |
| intro t ht |
| convert! (lowerDeficit_mass_deriv ht).sub ((gap_mass_deriv ht).const_mul δ) using 1 |
| · ext u |
| change u * (lowerDeficit u - δ * gap u) = u * lowerDeficit u - δ * (u * gap u) |
| ring |
| · dsimp [c] |
| ring |
| have hdc : ∀ t, 2 < t → HasDerivAt (fun u => u * c u) (-a (t - 1)) t := by |
| intro t ht |
| convert! (upperDeficit_mass_deriv ht).sub ((gap_mass_deriv ht).const_mul δ) using 1 |
| · ext u |
| change u * (upperDeficit u - δ * gap u) = u * upperDeficit u - δ * (u * gap u) |
| ring |
| · dsimp [a] |
| ring |
| have hla : Tendsto a atTop (𝓝 0) := by |
| simpa only [mul_zero, sub_zero] using lowerDeficit_tendsto_zero.sub (gap_tendsto_zero.const_mul δ) |
| have hlc : Tendsto c atTop (𝓝 0) := by |
| simpa only [mul_zero, sub_zero] using upperDeficit_tendsto_zero.sub (gap_tendsto_zero.const_mul δ) |
| exact PairedDeficitPositivity.paired_pos ha hc hai hci hsum hda hdc hla hlc s hs |
|
|
| theorem deficits_uniform_comparison : ∃ δ : ℝ, 0 < δ ∧ δ < 1 / 2 ∧ |
| ∀ s : ℝ, 1 ≤ s → |
| δ * gap s < lowerDeficit s ∧ lowerDeficit s < (1 - δ) * gap s ∧ |
| δ * gap s < upperDeficit s ∧ upperDeficit s < (1 - δ) * gap s := by |
| let δ : ℝ := min 1 (sieveA - 2) / (4 * sieveA) |
| have hδpos : 0 < δ := div_pos (lt_min zero_lt_one (sub_pos.mpr sieveA_gt_two)) |
| (mul_pos (by norm_num) sieveA_pos) |
| have heq : δ * sieveA = min 1 (sieveA - 2) / 4 := by |
| dsimp [δ] |
| field_simp [ne_of_gt sieveA_pos] |
| have hδA : δ * sieveA < 1 := by rw [heq]; linarith [min_le_left (1 : ℝ) (sieveA - 2)] |
| have hδA2 : δ * sieveA < sieveA - 2 := by |
| rw [heq] |
| linarith [min_le_right (1 : ℝ) (sieveA - 2), sieveA_gt_two] |
| have hhalf : δ < 1 / 2 := by nlinarith [sieveA_gt_two] |
| refine ⟨δ, hδpos, hhalf, ?_⟩ |
| intro s hs |
| obtain ⟨ha, hc⟩ := deficit_offsets_pos hhalf hδA hδA2 hs |
| have hsum := deficit_sum s |
| constructor |
| · linarith |
| constructor |
| · nlinarith |
| constructor |
| · linarith |
| · nlinarith |
|
|
| end NumberTheoryLean.NormalizedDeficits |
|
|
| end |
|
|
| section |
|
|
| namespace NumberTheoryLean.DeficitFutureIntegrals |
|
|
| open Filter Set MeasureTheory |
| open scoped Topology |
| open LinearSieveFunctions BuchstabBridge NormalizedDeficits |
|
|
| theorem future_integral_of_mass {a b : ℝ → ℝ} {base : ℝ} (hbase : 1 ≤ base) |
| (ha : ContinuousOn a (Ioi 0)) |
| (hb : ∀ t, base - 1 < t → 0 ≤ b t) |
| (hd : ∀ t, base < t → HasDerivAt (fun u => u * a u) (-b (t - 1)) t) |
| (hl : Tendsto (fun t => t * a t) atTop (𝓝 0)) |
| {s : ℝ} (hs : base ≤ s) : |
| IntegrableOn b (Ioi (s - 1)) ∧ s * a s = ∫ t in Ioi (s - 1), b t := by |
| let H : ℝ → ℝ := fun t => (t + 1) * a (t + 1) |
| have hs0 : 0 < s := lt_of_lt_of_le (by linarith : 0 < base) hs |
| have hc : ContinuousWithinAt H (Ici (s - 1)) (s - 1) := by |
| have hm : ContinuousAt (fun t => t * a t) s := |
| continuousAt_id.mul (ha.continuousAt (Ioi_mem_nhds hs0)) |
| have hm' : ContinuousAt (fun t => t * a t) ((s - 1) + 1) := by simpa only [sub_add_cancel] using hm |
| have hc' := hm'.comp (f := fun t : ℝ => t + 1) (x := s - 1) |
| ((continuous_id.add continuous_const : Continuous (fun t : ℝ => t + 1)).continuousAt) |
| convert! hc'.continuousWithinAt using 1 |
| have hder : ∀ t ∈ Ioi (s - 1), HasDerivAt H (-b t) t := by |
| intro t ht |
| have hdt := (hd (t + 1) (by change s - 1 < t at ht; linarith)).comp t |
| ((hasDerivAt_id t).add_const 1) |
| convert! hdt using 1 |
| simp |
| have hnonpos : ∀ t ∈ Ioi (s - 1), -b t ≤ 0 := by |
| intro t ht |
| exact neg_nonpos.mpr (hb t (by change s - 1 < t at ht; linarith)) |
| have hshift : Tendsto (fun t : ℝ => t + 1) atTop atTop := by |
| apply tendsto_atTop.mpr |
| intro v |
| exact eventually_atTop.mpr ⟨v - 1, fun t ht => by linarith⟩ |
| have hlim : Tendsto H atTop (𝓝 0) := hl.comp hshift |
| have hint := integrableOn_Ioi_deriv_of_nonpos hc hder hnonpos hlim |
| have heq := integral_Ioi_of_hasDerivAt_of_nonpos hc hder hnonpos hlim |
| constructor |
| · exact integrable_neg_iff.mp hint |
| · simp only [integral_neg, H, sub_add_cancel, zero_sub] at heq |
| linarith |
|
|
| theorem deficit_mass_bounds {s : ℝ} (hs : 1 ≤ s) : |
| (0 ≤ s * lowerDeficit s ∧ s * lowerDeficit s ≤ sieveA * Dickman.rho (s - 1)) ∧ |
| (0 ≤ s * upperDeficit s ∧ s * upperDeficit s ≤ sieveA * Dickman.rho (s - 1)) := by |
| obtain ⟨ha, hc⟩ := deficits_pos hs |
| have hsum := deficit_sum s |
| have hg : s * gap s = sieveA * Dickman.rho (s - 1) := by |
| rw [gap_eq_rho hs] |
| field_simp |
| have hs0 : 0 < s := lt_of_lt_of_le zero_lt_one hs |
| have hbA : s * lowerDeficit s ≤ s * gap s := |
| mul_le_mul_of_nonneg_left (by linarith) hs0.le |
| have hbC : s * upperDeficit s ≤ s * gap s := |
| mul_le_mul_of_nonneg_left (by linarith) hs0.le |
| exact ⟨⟨mul_nonneg hs0.le ha.le, by simpa only [hg] using hbA⟩, |
| ⟨mul_nonneg hs0.le hc.le, by simpa only [hg] using hbC⟩⟩ |
|
|
| theorem lowerDeficit_mass_tendsto_zero : |
| Tendsto (fun s => s * lowerDeficit s) atTop (𝓝 0) := by |
| have hshift : Tendsto (fun s : ℝ => s - 1) atTop atTop := by |
| apply tendsto_atTop.mpr |
| intro v |
| exact eventually_atTop.mpr ⟨v + 1, fun s hs => by linarith⟩ |
| have hlim := (DickmanDecay.rho_tendsto_zero.comp hshift).const_mul sieveA |
| simp only [mul_zero] at hlim |
| apply squeeze_zero' ?_ ?_ hlim |
| · filter_upwards [eventually_ge_atTop (1 : ℝ)] with s hs |
| exact (deficit_mass_bounds hs).1.1 |
| · filter_upwards [eventually_ge_atTop (1 : ℝ)] with s hs |
| exact (deficit_mass_bounds hs).1.2 |
| |
| theorem upperDeficit_mass_tendsto_zero : |
| Tendsto (fun s => s * upperDeficit s) atTop (𝓝 0) := by |
| have hshift : Tendsto (fun s : ℝ => s - 1) atTop atTop := by |
| apply tendsto_atTop.mpr |
| intro v |
| exact eventually_atTop.mpr ⟨v + 1, fun s hs => by linarith⟩ |
| have hlim := (DickmanDecay.rho_tendsto_zero.comp hshift).const_mul sieveA |
| simp only [mul_zero] at hlim |
| apply squeeze_zero' ?_ ?_ hlim |
| · filter_upwards [eventually_ge_atTop (1 : ℝ)] with s hs |
| exact (deficit_mass_bounds hs).2.1 |
| · filter_upwards [eventually_ge_atTop (1 : ℝ)] with s hs |
| exact (deficit_mass_bounds hs).2.2 |
|
|
| theorem upperDeficit_mass_deriv_of_one_lt {s : ℝ} (hs : 1 < s) : |
| HasDerivAt (fun t => t * upperDeficit t) (-lowerDeficit (s - 1)) s := by |
| by_cases hs2 : 2 < s |
| · exact upperDeficit_mass_deriv hs2 |
| · have heq : (fun t => t * upperDeficit t) =ᶠ[𝓝 s] (fun t => sieveA - t) := by |
| filter_upwards [Ioo_mem_nhds (show 0 < s by linarith) (show s < 3 by linarith)] with t ht |
| change t * (F sieveA t - 1) = sieveA - t |
| rw [F_initial sieveA ht.1 ht.2.le] |
| field_simp [ne_of_gt ht.1] |
| have h := ((hasDerivAt_const s sieveA).sub (hasDerivAt_id s)).congr_of_eventuallyEq heq |
| convert! h using 1 |
| simp only [lowerDeficit, f_initial sieveA (s := s - 1) (by linarith), sub_zero, zero_sub] |
|
|
| theorem lowerDeficit_nonneg (s : ℝ) : 0 ≤ lowerDeficit s := by |
| by_cases hs : 1 ≤ s |
| · exact (deficits_pos hs).1.le |
| · simp only [lowerDeficit, f_initial sieveA (by linarith : s ≤ 2), sub_zero, zero_le_one] |
|
|
| theorem lower_future_integral {s : ℝ} (hs : 2 ≤ s) : |
| IntegrableOn upperDeficit (Ioi (s - 1)) ∧ |
| s * lowerDeficit s = ∫ t in Ioi (s - 1), upperDeficit t := by |
| exact future_integral_of_mass (a := lowerDeficit) (b := upperDeficit) (base := 2) (by norm_num) |
| lowerDeficit_continuous.continuousOn (fun t ht => (deficits_pos (s := t) (by linarith)).2.le) |
| (fun _ ht => lowerDeficit_mass_deriv ht) lowerDeficit_mass_tendsto_zero hs |
|
|
| theorem upper_future_integral {s : ℝ} (hs : 1 ≤ s) : |
| IntegrableOn lowerDeficit (Ioi (s - 1)) ∧ |
| s * upperDeficit s = ∫ t in Ioi (s - 1), lowerDeficit t := by |
| exact future_integral_of_mass (a := upperDeficit) (b := lowerDeficit) (base := 1) le_rfl |
| ((F_continuousOn sieveA).sub continuousOn_const) (fun t _ => lowerDeficit_nonneg t) |
| (fun _ ht => upperDeficit_mass_deriv_of_one_lt ht) upperDeficit_mass_tendsto_zero hs |
|
|
| end NumberTheoryLean.DeficitFutureIntegrals |
|
|
| end |
|
|
| section |
|
|
| namespace NumberTheoryLean.DerivativeWeights |
|
|
| open Filter Set MeasureTheory |
| open scoped Topology |
| open LinearSieveFunctions BuchstabBridge NormalizedDeficits DeficitFutureIntegrals |
|
|
| noncomputable def evenInitial (s : ℝ) : ℝ := |
| sieveA * (s / (s - 1) - Real.log (s - 1)) |
|
|
| noncomputable def phiEven (s : ℝ) : ℝ := |
| if s ≤ 2 then evenInitial s else s * (lowerDeficit s + upperDeficit (s - 1)) |
|
|
| noncomputable def phiOdd (s : ℝ) : ℝ := |
| if s ≤ 3 then sieveA else s * (upperDeficit s + lowerDeficit (s - 1)) |
|
|
| noncomputable def W (t : ℝ) : ℝ := (t + 1) / t ^ 2 |
|
|
| theorem phiOdd_initial {s : ℝ} (hs : s ≤ 3) : phiOdd s = sieveA := by |
| simp only [phiOdd, ite_eq_left hs] |
|
|
| theorem phiEven_initial {s : ℝ} (hs : s ≤ 2) : phiEven s = evenInitial s := by |
| simp only [phiEven, ite_eq_left hs] |
|
|
| theorem phiEven_eq_deficits {s : ℝ} (hs : 2 ≤ s) : |
| phiEven s = s * (lowerDeficit s + upperDeficit (s - 1)) := by |
| rcases eq_or_lt_of_le hs with h | h |
| · subst s |
| norm_num [phiEven, evenInitial, lowerDeficit, upperDeficit, |
| f_initial, F_initial sieveA (s := 1) (by norm_num) (by norm_num)] |
| ring |
| · exact ite_eq_right (not_le_of_gt h) |
|
|
| theorem phiOdd_eq_deficits {s : ℝ} (hs : 1 ≤ s) : |
| phiOdd s = s * (upperDeficit s + lowerDeficit (s - 1)) := by |
| by_cases hs3 : s ≤ 3 |
| · rw [phiOdd_initial hs3] |
| simp only [upperDeficit, lowerDeficit, f_initial sieveA (s := s - 1) (by linarith), |
| F_initial sieveA (s := s) (by linarith) hs3] |
| field_simp |
| ring |
| · exact ite_eq_right hs3 |
|
|
| theorem phiEven_explicit {s : ℝ} (hs : 1 < s) (hs4 : s ≤ 4) : |
| phiEven s = evenInitial s := by |
| by_cases hs2 : s ≤ 2 |
| · exact phiEven_initial hs2 |
| · rw [phiEven_eq_deficits (by linarith)] |
| unfold lowerDeficit upperDeficit evenInitial |
| rw [f_first_formula sieveA (by linarith) hs4, |
| F_initial sieveA (s := s - 1) (by linarith) (by linarith)] |
| field_simp |
| ring |
|
|
| theorem hasDerivAt_of_mass {g : ℝ → ℝ} {s v : ℝ} (hs : s ≠ 0) |
| (hm : HasDerivAt (fun t => t * g t) v s) : |
| HasDerivAt g ((v - g s) / s) s := by |
| have heq : g =ᶠ[𝓝 s] (fun t => (t * g t) / t) := by |
| filter_upwards [eventually_ne_nhds hs] with t ht |
| field_simp |
| have h := (hm.div (hasDerivAt_id s) hs).congr_of_eventuallyEq heq |
| convert! h using 1 |
| simp only [id_eq, mul_one] |
| field_simp |
|
|
| theorem lowerDeficit_hasDerivAt {s : ℝ} (hs : 2 < s) : |
| HasDerivAt lowerDeficit (-phiEven s / s ^ 2) s := by |
| have h := hasDerivAt_of_mass (by linarith : s ≠ 0) (lowerDeficit_mass_deriv hs) |
| convert! h using 1 |
| rw [phiEven_eq_deficits hs.le] |
| field_simp |
| ring |
|
|
| theorem upperDeficit_hasDerivAt {s : ℝ} (hs : 1 < s) : |
| HasDerivAt upperDeficit (-phiOdd s / s ^ 2) s := by |
| have h := hasDerivAt_of_mass (by linarith : s ≠ 0) (upperDeficit_mass_deriv_of_one_lt hs) |
| convert! h using 1 |
| rw [phiOdd_eq_deficits hs.le] |
| field_simp |
| ring |
|
|
| theorem evenInitial_hasDerivAt {s : ℝ} (hs : 1 < s) : |
| HasDerivAt evenInitial (-s / (s - 1) ^ 2 * sieveA) s := by |
| have hn : s - 1 ≠ 0 := by linarith |
| have h := (((hasDerivAt_id s).div ((hasDerivAt_id s).sub_const 1) hn).sub |
| (((hasDerivAt_id s).sub_const 1).log hn)).const_mul sieveA |
| convert! h using 1 |
| simp only [id_eq, one_mul, mul_one] |
| field_simp |
| ring |
|
|
| theorem phiEven_hasDerivAt {s : ℝ} (hs : 1 < s) : |
| HasDerivAt phiEven (-s / (s - 1) ^ 2 * phiOdd (s - 1)) s := by |
| by_cases hs3 : s < 3 |
| · have heq : phiEven =ᶠ[𝓝 s] evenInitial := by |
| filter_upwards [Ioo_mem_nhds hs hs3] with t ht |
| exact phiEven_explicit ht.1 (by linarith [ht.2]) |
| rw [phiOdd_initial (s := s - 1) (by linarith)] |
| exact (evenInitial_hasDerivAt hs).congr_of_eventuallyEq heq |
| · have hs2 : 2 < s := by linarith |
| have hU := (upperDeficit_hasDerivAt (s := s - 1) (by linarith)).comp s |
| ((hasDerivAt_id s).sub_const 1) |
| have h := (hasDerivAt_id s).mul ((lowerDeficit_hasDerivAt hs2).add hU) |
| have heq : phiEven =ᶠ[𝓝 s] (fun t => t * (lowerDeficit t + upperDeficit (t - 1))) := by |
| filter_upwards [Ioi_mem_nhds hs2] with t ht |
| exact phiEven_eq_deficits ht.le |
| have hd := h.congr_of_eventuallyEq heq |
| convert! hd using 1 |
| simp only [id_eq, Pi.add_apply, Function.comp_apply, one_mul, mul_one] |
| rw [phiEven_eq_deficits hs2.le] |
| field_simp |
| ring |
|
|
| theorem phiOdd_hasDerivAt {s : ℝ} (hs : 3 < s) : |
| HasDerivAt phiOdd (-s / (s - 1) ^ 2 * phiEven (s - 1)) s := by |
| have hA := (lowerDeficit_hasDerivAt (s := s - 1) (by linarith)).comp s |
| ((hasDerivAt_id s).sub_const 1) |
| have h := (hasDerivAt_id s).mul ((upperDeficit_hasDerivAt (by linarith : 1 < s)).add hA) |
| have heq : phiOdd =ᶠ[𝓝 s] (fun t => t * (upperDeficit t + lowerDeficit (t - 1))) := by |
| filter_upwards [Ioi_mem_nhds (show 1 < s by linarith)] with t ht |
| exact phiOdd_eq_deficits ht.le |
| have hd := h.congr_of_eventuallyEq heq |
| convert! hd using 1 |
| simp only [id_eq, Pi.add_apply, Function.comp_apply, one_mul, mul_one] |
| rw [phiOdd_eq_deficits (by linarith)] |
| field_simp |
| ring |
|
|
| theorem phiEven_pos {s : ℝ} (hs : 1 < s) : 0 < phiEven s := by |
| by_cases hs2 : 2 ≤ s |
| · rw [phiEven_eq_deficits hs2] |
| exact mul_pos (by linarith) (add_pos (deficits_pos (by linarith)).1 |
| (deficits_pos (s := s - 1) (by linarith)).2) |
| · rw [phiEven_initial (by linarith)] |
| unfold evenInitial |
| have hlog : Real.log (s - 1) < 0 := Real.log_neg (by linarith) (by linarith) |
| exact mul_pos sieveA_pos (sub_pos.mpr (lt_trans hlog (div_pos (by linarith) (by linarith)))) |
|
|
| theorem phiOdd_pos (s : ℝ) : 0 < phiOdd s := by |
| by_cases hs3 : s ≤ 3 |
| · rw [phiOdd_initial hs3] |
| exact sieveA_pos |
| · rw [phiOdd_eq_deficits (by linarith)] |
| exact mul_pos (by linarith) (add_pos (deficits_pos (by linarith)).2 |
| (deficits_pos (s := s - 1) (by linarith)).1) |
|
|
| theorem phiEven_continuousAt {s : ℝ} (hs : 1 < s) : ContinuousAt phiEven s := |
| (phiEven_hasDerivAt hs).continuousAt |
|
|
| theorem phiOdd_continuous : Continuous phiOdd := by |
| apply continuous_iff_continuousAt.mpr |
| intro s |
| by_cases hs3 : s < 3 |
| · have heq : phiOdd =ᶠ[𝓝 s] (fun _ => sieveA) := by |
| filter_upwards [Iio_mem_nhds hs3] with t ht |
| exact phiOdd_initial ht.le |
| exact continuousAt_const.congr_of_eventuallyEq heq |
| · have heq : phiOdd =ᶠ[𝓝 s] (fun t => t * (upperDeficit t + lowerDeficit (t - 1))) := by |
| filter_upwards [Ioi_mem_nhds (show 1 < s by linarith)] with t ht |
| exact phiOdd_eq_deficits ht.le |
| have hcU : ContinuousAt upperDeficit s := |
| (F_continuousAt sieveA (by linarith)).sub continuousAt_const |
| have hcA : ContinuousAt (fun t => lowerDeficit (t - 1)) s := |
| (lowerDeficit_continuous.comp (continuous_id.sub continuous_const)).continuousAt |
| exact (continuousAt_id.mul (hcU.add hcA)).congr_of_eventuallyEq heq |
|
|
| theorem f_hasDerivAt {s : ℝ} (hs : 2 < s) : |
| HasDerivAt (f sieveA) (phiEven s / s ^ 2) s := by |
| convert! (lowerDeficit_hasDerivAt hs).const_sub 1 using 1 |
| · ext t |
| change f sieveA t = 1 - (1 - f sieveA t) |
| ring |
| · ring |
|
|
| theorem F_hasDerivAt {s : ℝ} (hs : 0 < s) : |
| HasDerivAt (F sieveA) (-phiOdd s / s ^ 2) s := by |
| by_cases hs1 : 1 < s |
| · convert! (upperDeficit_hasDerivAt hs1).add_const 1 using 1 |
| ext t |
| change F sieveA t = (F sieveA t - 1) + 1 |
| ring |
| · have heq : F sieveA =ᶠ[𝓝 s] (fun t => sieveA / t) := by |
| filter_upwards [Ioo_mem_nhds hs (show s < 3 by linarith)] with t ht |
| exact F_initial sieveA ht.1 ht.2.le |
| have hd := ((hasDerivAt_const s sieveA).div (hasDerivAt_id s) (ne_of_gt hs)).congr_of_eventuallyEq heq |
| convert! hd using 1 |
| rw [phiOdd_initial (by linarith)] |
| simp only [id_eq, zero_mul, zero_sub, mul_one] |
|
|
| theorem startingExtension_hasDerivAt {s : ℝ} (hs : 1 < s) : |
| HasDerivAt (startingExtension sieveA) (evenInitial s / s ^ 2) s := by |
| have hn : s - 1 ≠ 0 := by linarith |
| have hd := ((((hasDerivAt_id s).sub_const 1).log hn).const_mul sieveA).div |
| (hasDerivAt_id s) (by linarith : s ≠ 0) |
| convert! hd using 1 |
| simp only [evenInitial, id_eq, mul_one] |
| field_simp |
|
|
| theorem f_right_derivative_two : |
| HasDerivWithinAt (f sieveA) (phiEven 2 / 2 ^ 2) (Ici 2) 2 := by |
| rw [phiEven_initial le_rfl] |
| apply (startingExtension_hasDerivAt (s := 2) (by norm_num)).hasDerivWithinAt.congr_of_eventuallyEq_of_mem |
| · filter_upwards [self_mem_nhdsWithin, |
| (nhdsWithin_le_nhds (Iio_mem_nhds (show (2 : ℝ) < 4 by norm_num)))] with t ht ht4 |
| exact (startingExtension_eq_f sieveA ht ht4.le).symm |
| · exact self_mem_Ici |
|
|
| theorem phiEven_eq_deriv {s : ℝ} (hs : 2 < s) : |
| phiEven s = s ^ 2 * deriv (f sieveA) s := by |
| rw [(f_hasDerivAt hs).deriv] |
| field_simp |
|
|
| theorem phiOdd_eq_deriv {s : ℝ} (hs : 0 < s) : |
| phiOdd s = -s ^ 2 * deriv (F sieveA) s := by |
| rw [(F_hasDerivAt hs).deriv] |
| field_simp |
|
|
| theorem phiEven_eq_right_deriv_two : |
| phiEven 2 = 2 ^ 2 * derivWithin (f sieveA) (Ici 2) 2 := by |
| rw [f_right_derivative_two.derivWithin (uniqueDiffWithinAt_Ici (2 : ℝ))] |
| ring |
|
|
| theorem phiEven_strictAntiOn : StrictAntiOn phiEven (Ioi 1) := by |
| apply strictAntiOn_of_deriv_neg (convex_Ioi 1) |
| · intro s hs |
| exact (phiEven_continuousAt hs).continuousWithinAt |
| · intro s hs |
| rw [interior_Ioi] at hs |
| change 1 < s at hs |
| rw [(phiEven_hasDerivAt hs).deriv] |
| exact mul_neg_of_neg_of_pos (div_neg_of_neg_of_pos (by linarith) |
| (sq_pos_of_pos (by linarith))) (phiOdd_pos _) |
|
|
| theorem phiOdd_antitone : Antitone phiOdd := by |
| have htail : AntitoneOn phiOdd (Ici 3) := by |
| apply antitoneOn_of_deriv_nonpos (convex_Ici 3) phiOdd_continuous.continuousOn |
| · intro s hs |
| rw [interior_Ici] at hs |
| exact (phiOdd_hasDerivAt hs).differentiableAt.differentiableWithinAt |
| · intro s hs |
| rw [interior_Ici] at hs |
| change 3 < s at hs |
| rw [(phiOdd_hasDerivAt hs).deriv] |
| exact mul_nonpos_of_nonpos_of_nonneg (div_nonpos_of_nonpos_of_nonneg (by linarith) (sq_nonneg _)) |
| (phiEven_pos (by linarith)).le |
| intro x y hxy |
| by_cases hy : y ≤ 3 |
| · rw [phiOdd_initial hy, phiOdd_initial (le_trans hxy hy)] |
| · by_cases hx : x ≤ 3 |
| · rw [phiOdd_initial hx, ← phiOdd_initial (s := 3) le_rfl] |
| exact htail (by simp) (le_of_not_ge hy) (le_of_not_ge hy) |
| · exact htail (le_of_not_ge hx) (le_of_not_ge hy) hxy |
|
|
| end NumberTheoryLean.DerivativeWeights |
|
|
| end |
|
|
| section |
|
|
| namespace NumberTheoryLean.WeightFutureIntegrals |
|
|
| open Filter Set MeasureTheory |
| open scoped Topology |
| open NormalizedDeficits DeficitFutureIntegrals DerivativeWeights |
|
|
| theorem phiEven_tendsto_zero : Tendsto phiEven atTop (𝓝 0) := by |
| have hshift : Tendsto (fun s : ℝ => s - 1) atTop atTop := by |
| apply tendsto_atTop.mpr |
| intro v |
| exact eventually_atTop.mpr ⟨v + 1, fun s hs => by linarith⟩ |
| have hl := lowerDeficit_mass_tendsto_zero.add |
| ((upperDeficit_mass_tendsto_zero.comp hshift).add (upperDeficit_tendsto_zero.comp hshift)) |
| simp only [add_zero] at hl |
| apply hl.congr' |
| filter_upwards [eventually_ge_atTop (2 : ℝ)] with s hs |
| rw [phiEven_eq_deficits hs] |
| simp only [Function.comp_apply] |
| ring |
| |
| theorem phiOdd_tendsto_zero : Tendsto phiOdd atTop (𝓝 0) := by |
| have hshift : Tendsto (fun s : ℝ => s - 1) atTop atTop := by |
| apply tendsto_atTop.mpr |
| intro v |
| exact eventually_atTop.mpr ⟨v + 1, fun s hs => by linarith⟩ |
| have hl := upperDeficit_mass_tendsto_zero.add |
| ((lowerDeficit_mass_tendsto_zero.comp hshift).add (lowerDeficit_tendsto_zero.comp hshift)) |
| simp only [add_zero] at hl |
| apply hl.congr' |
| filter_upwards [eventually_ge_atTop (1 : ℝ)] with s hs |
| rw [phiOdd_eq_deficits hs] |
| simp only [Function.comp_apply] |
| ring |
|
|
| theorem W_pos {t : ℝ} (ht : 0 < t) : 0 < W t := by |
| unfold W |
| positivity |
|
|
| theorem future_integral_of_shifted_deriv {P B : ℝ → ℝ} {s : ℝ} |
| (hc : ContinuousAt P s) |
| (hd : ∀ t, s < t → HasDerivAt P (-B (t - 1)) t) |
| (hb : ∀ t, s - 1 < t → 0 ≤ B t) |
| (hl : Tendsto P atTop (𝓝 0)) : |
| IntegrableOn B (Ioi (s - 1)) ∧ P s = ∫ t in Ioi (s - 1), B t := by |
| let H : ℝ → ℝ := fun t => P (t + 1) |
| have hc' : ContinuousWithinAt H (Ici (s - 1)) (s - 1) := by |
| have hcs : ContinuousAt P ((s - 1) + 1) := by simpa only [sub_add_cancel] using hc |
| exact (hcs.comp (f := fun t : ℝ => t + 1) (x := s - 1) |
| (continuous_id.add continuous_const).continuousAt).continuousWithinAt |
| have hder : ∀ t ∈ Ioi (s - 1), HasDerivAt H (-B t) t := by |
| intro t ht |
| have h := (hd (t + 1) (by change s - 1 < t at ht; linarith)).comp t |
| ((hasDerivAt_id t).add_const 1) |
| convert! h using 1 |
| simp |
| have hn : ∀ t ∈ Ioi (s - 1), -B t ≤ 0 := fun t ht => neg_nonpos.mpr (hb t ht) |
| have hshift : Tendsto (fun t : ℝ => t + 1) atTop atTop := by |
| apply tendsto_atTop.mpr |
| intro v |
| exact eventually_atTop.mpr ⟨v - 1, fun t ht => by linarith⟩ |
| have hlim : Tendsto H atTop (𝓝 0) := hl.comp hshift |
| have hint := integrableOn_Ioi_deriv_of_nonpos hc' hder hn hlim |
| have heq := integral_Ioi_of_hasDerivAt_of_nonpos hc' hder hn hlim |
| constructor |
| · exact integrable_neg_iff.mp hint |
| · simp only [integral_neg, H, sub_add_cancel, zero_sub] at heq |
| linarith |
| |
| theorem phiEven_future_integral {s : ℝ} (hs : 1 < s) : |
| IntegrableOn (fun t => W t * phiOdd t) (Ioi (s - 1)) ∧ |
| phiEven s = ∫ t in Ioi (s - 1), W t * phiOdd t := by |
| apply future_integral_of_shifted_deriv (phiEven_continuousAt hs) ?_ ?_ phiEven_tendsto_zero |
| · intro t ht |
| convert! phiEven_hasDerivAt (lt_trans hs ht) using 1 |
| simp only [W, sub_add_cancel] |
| ring |
| · intro t ht |
| exact (mul_pos (W_pos (by linarith)) (phiOdd_pos t)).le |
| |
| theorem phiOdd_future_integral (s : ℝ) : |
| IntegrableOn (fun t => W t * phiEven t) (Ioi (max 2 (s - 1))) ∧ |
| phiOdd s = ∫ t in Ioi (max 2 (s - 1)), W t * phiEven t := by |
| have hc := phiOdd_continuous.continuousAt (x := max 3 s) |
| have hder : ∀ t, max 3 s < t → |
| HasDerivAt phiOdd (-(W (t - 1) * phiEven (t - 1))) t := by |
| intro t ht |
| convert! phiOdd_hasDerivAt (lt_of_le_of_lt (le_max_left _ _) ht) using 1 |
| simp only [W, sub_add_cancel] |
| ring |
| have hn : ∀ t, max 3 s - 1 < t → 0 ≤ W t * phiEven t := by |
| intro t ht |
| have ht1 : 1 < t := by linarith [le_max_left (3 : ℝ) s] |
| exact (mul_pos (W_pos (by linarith)) (phiEven_pos ht1)).le |
| have h := future_integral_of_shifted_deriv hc hder hn phiOdd_tendsto_zero |
| have hmax : max 3 s - 1 = max 2 (s - 1) := by |
| by_cases hs : s ≤ 3 |
| · rw [max_eq_left hs, max_eq_left (by linarith)] |
| norm_num |
| · rw [max_eq_right (le_of_not_ge hs), max_eq_right (by linarith)] |
| have heq : phiOdd (max 3 s) = phiOdd s := by |
| by_cases hs : s ≤ 3 |
| · rw [max_eq_left hs, phiOdd_initial le_rfl, phiOdd_initial hs] |
| · rw [max_eq_right (le_of_not_ge hs)] |
| simpa only [hmax, heq] using h |
| |
| end NumberTheoryLean.WeightFutureIntegrals |
| |
| end |
| |
| end Erdos970 |
| |
| end OAI |
| |
