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aaf4b9f | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 56 57 58 59 60 61 62 63 64 65 | import OAI.NumberTheory.Ostmann.PrimeProgression.CountToHarmonicMain
namespace OAI
open scoped BigOperators
open Set MeasureTheory
namespace Ostmann.Arithmetic.PrimeProgression
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
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
end Ostmann.Arithmetic.PrimeProgression
end OAI
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