| import OAI.NumberTheory.Ostmann.PrimeProgression.CountToHarmonicMain |
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| namespace OAI |
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| open scoped BigOperators |
| open Set MeasureTheory |
| namespace Ostmann.Arithmetic.PrimeProgression |
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| lemma abs_inv_mul_le_of_abs_le_mul {x δ r : ℝ} (hx : 0 < x) (hr : |r| ≤ δ * x) : |
| |x⁻¹ * r| ≤ δ := by |
| rw [abs_mul, abs_of_pos (inv_pos.mpr hx)] |
| calc |
| x⁻¹ * |r| ≤ x⁻¹ * (δ * x) := mul_le_mul_of_nonneg_left hr (inv_nonneg.mpr hx.le) |
| _ = δ := by field_simp |
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| theorem reciprocal_count_error_bound (c : ℕ → ℝ) (F D : ℝ → ℝ) {A B δ : ℝ} |
| (hA : 0 < A) (hAB : A ≤ B) |
| (hF : ∀ t ∈ Icc A B, HasDerivAt F (D t) t) (hD : ContinuousOn D (Icc A B)) |
| (herror : ∀ t ∈ Icc A B, |cumulativeSum c t - F t| ≤ δ * t) : |
| |(∑ n ∈ Finset.Ioc ⌊A⌋₊ ⌊B⌋₊, (n : ℝ)⁻¹ * c n) - |
| (∫ t in A..B, t⁻¹ * D t)| ≤ (2 + Real.log B - Real.log A) * δ := by |
| have hB := hA.trans_le hAB |
| have hiC : IntervalIntegrable (fun t => (t ^ 2)⁻¹ * cumulativeSum c t) volume A B := |
| (intervalIntegrable_iff_integrableOn_Icc_of_le hAB).mpr |
| (cumulativeSum_inv_sq_integrable c hA) |
| have hiF : IntervalIntegrable (fun t => (t ^ 2)⁻¹ * F t) volume A B := |
| ((continuousOn_inv_sq hA).mul (continuousOn_of_hasDerivAt_Icc F D hF)).intervalIntegrable_of_Icc hAB |
| have hiK : IntervalIntegrable (fun t => (t ^ 2)⁻¹ * (cumulativeSum c t - F t)) volume A B := by |
| simp_rw [mul_sub] |
| exact hiC.sub hiF |
| have hiInv : IntervalIntegrable (fun t : ℝ => δ * t⁻¹) volume A B := |
| ((continuousOn_inv_Icc hA).const_mul δ).intervalIntegrable_of_Icc hAB |
| have hpoint (t : ℝ) (ht : t ∈ Icc A B) : |
| |(t ^ 2)⁻¹ * (cumulativeSum c t - F t)| ≤ δ * t⁻¹ := by |
| have htpos : 0 < t := hA.trans_le ht.1 |
| rw [abs_mul, abs_of_nonneg (inv_nonneg.mpr (sq_nonneg t))] |
| calc |
| (t ^ 2)⁻¹ * |cumulativeSum c t - F t| ≤ (t ^ 2)⁻¹ * (δ * t) := |
| mul_le_mul_of_nonneg_left (herror t ht) (inv_nonneg.mpr (sq_nonneg t)) |
| _ = δ * t⁻¹ := by field_simp |
| have hIntegral : |∫ t in A..B, (t ^ 2)⁻¹ * (cumulativeSum c t - F t)| ≤ |
| δ * (Real.log B - Real.log A) := by |
| calc |
| _ ≤ ∫ t in A..B, |(t ^ 2)⁻¹ * (cumulativeSum c t - F t)| := |
| intervalIntegral.abs_integral_le_integral_abs hAB |
| _ ≤ ∫ t in A..B, δ * t⁻¹ := |
| intervalIntegral.integral_mono_on hAB hiK.abs hiInv hpoint |
| _ = _ := by |
| rw [intervalIntegral.integral_const_mul, integral_inv_of_pos hA hB, |
| Real.log_div hB.ne' hA.ne'] |
| have hEA := abs_inv_mul_le_of_abs_le_mul hA (herror A ⟨le_rfl, hAB⟩) |
| have hEB := abs_inv_mul_le_of_abs_le_mul hB (herror B ⟨hAB, le_rfl⟩) |
| rw [reciprocal_count_error_identity c F D hA hAB hF hD] |
| calc |
| _ ≤ |B⁻¹ * (cumulativeSum c B - F B)| + |A⁻¹ * (cumulativeSum c A - F A)| + |
| |∫ t in A..B, (t ^ 2)⁻¹ * (cumulativeSum c t - F t)| := |
| (abs_add_le _ _).trans (add_le_add (abs_sub _ _) le_rfl) |
| _ ≤ δ + δ + δ * (Real.log B - Real.log A) := |
| add_le_add (add_le_add hEB hEA) hIntegral |
| _ = _ := by ring |
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| end Ostmann.Arithmetic.PrimeProgression |
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| end OAI |
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