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180bc03 | 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 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 | import OAI.NumberTheory.TwoPointCorrelations.HalaszPrimeConvolution
namespace OAI
/-! A complex-valued approximation, not just a norm bound, permits a second
application of the prime convolution inside the first one. -/
namespace TwoPointCorrelations
open Finset
lemma halasz_prime_log_approximation (f : ℕ → ℂ) (hf : OneBounded f)
(hmul : ∀ m n : ℕ, 0 < m → 0 < n → f (m * n) = f m * f n)
(M : ℕ) (hM : 1 ≤ M) :
‖(∑ n ∈ Icc 1 M, f n) * (Real.log (M : ℝ) : ℂ) -
halaszPrimeConvolution f M‖ ≤ (halaszPrimePowerConstant + 1) * (M : ℝ) := by
have he := halasz_logarithmic_sum_error f hf M hM
have hp := halasz_log_sum_prime_error f hf hmul M
have ht := norm_add_le
((∑ n ∈ Icc 1 M, f n) * (Real.log (M : ℝ) : ℂ) -
∑ n ∈ Icc 1 M, f n * (Real.log (n : ℝ) : ℂ))
((∑ n ∈ Icc 1 M, f n * (Real.log (n : ℝ) : ℂ)) -
halaszPrimeConvolution f M)
rw [sub_add_sub_cancel] at ht
linarith
lemma halasz_prime_inner_approximation (f : ℕ → ℂ) (hf : OneBounded f)
(hmul : ∀ m n : ℕ, 0 < m → 0 < n → f (m * n) = f m * f n)
(M : ℕ) (hM : 2 ≤ M) :
‖(∑ n ∈ Icc 1 M, f n) - halaszPrimeConvolution f M / (Real.log (M : ℝ) : ℂ)‖ ≤
(halaszPrimePowerConstant + 1) * (M : ℝ) / Real.log (M : ℝ) := by
have hlog : 0 < Real.log (M : ℝ) := Real.log_pos (by exact_mod_cast hM)
have hc : (Real.log (M : ℝ) : ℂ) ≠ 0 := by exact_mod_cast ne_of_gt hlog
have he : (∑ n ∈ Icc 1 M, f n) -
halaszPrimeConvolution f M / (Real.log (M : ℝ) : ℂ) =
((∑ n ∈ Icc 1 M, f n) * (Real.log (M : ℝ) : ℂ) -
halaszPrimeConvolution f M) / (Real.log (M : ℝ) : ℂ) := by
field_simp
rw [he, norm_div, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hlog]
exact div_le_div_of_nonneg_right (halasz_prime_log_approximation f hf hmul M (by omega))
(le_of_lt hlog)
noncomputable def halaszInnerErrorConstant : ℝ :=
halaszPrimePowerConstant + 1 + Real.log 2
lemma halaszInnerErrorConstant_nonneg : 0 ≤ halaszInnerErrorConstant := by
unfold halaszInnerErrorConstant
have hC := halaszPrimePowerConstant_nonneg
have h2 : 0 ≤ Real.log 2 := Real.log_nonneg (by norm_num)
linarith
/-- The logarithm may use the real cutoff, while the sum and convolution
retain the exact natural cutoff. Their rounding error is uniformly O(x). -/
lemma halasz_real_log_approximation (f : ℕ → ℂ) (hf : OneBounded f)
(hmul : ∀ m n : ℕ, 0 < m → 0 < n → f (m * n) = f m * f n)
(M : ℕ) (hM : 1 ≤ M) {x : ℝ} (hMx : (M : ℝ) ≤ x) (hxM : x ≤ 2 * M) :
‖(∑ n ∈ Icc 1 M, f n) * (Real.log x : ℂ) -
halaszPrimeConvolution f M‖ ≤ halaszInnerErrorConstant * x := by
have hM0 : (0 : ℝ) < M := by exact_mod_cast hM
have hx0 : 0 < x := hM0.trans_le hMx
have hlog0 : 0 ≤ Real.log x - Real.log (M : ℝ) :=
sub_nonneg.mpr (Real.log_le_log hM0 hMx)
have hlog2 : Real.log x - Real.log (M : ℝ) ≤ Real.log 2 := by
have h := Real.log_le_log hx0 hxM
rw [Real.log_mul (by norm_num) hM0.ne'] at h
linarith
have he : (∑ n ∈ Icc 1 M, f n) * (Real.log x : ℂ) -
halaszPrimeConvolution f M =
(∑ n ∈ Icc 1 M, f n) * ((Real.log x - Real.log (M : ℝ) : ℝ) : ℂ) +
((∑ n ∈ Icc 1 M, f n) * (Real.log (M : ℝ) : ℂ) -
halaszPrimeConvolution f M) := by push_cast; ring
rw [he]
calc
_ ≤ ‖(∑ n ∈ Icc 1 M, f n) * ((Real.log x - Real.log (M : ℝ) : ℝ) : ℂ)‖ +
‖(∑ n ∈ Icc 1 M, f n) * (Real.log (M : ℝ) : ℂ) -
halaszPrimeConvolution f M‖ := norm_add_le _ _
_ ≤ (M : ℝ) * Real.log 2 + (halaszPrimePowerConstant + 1) * M := by
apply add_le_add _ (halasz_prime_log_approximation f hf hmul M hM)
rw [norm_mul, Complex.norm_real, Real.norm_eq_abs, abs_of_nonneg hlog0]
exact mul_le_mul (halasz_bounded_prefix_norm f hf M) hlog2 hlog0 (Nat.cast_nonneg _)
_ = halaszInnerErrorConstant * (M : ℝ) := by unfold halaszInnerErrorConstant; ring
_ ≤ _ := mul_le_mul_of_nonneg_left hMx halaszInnerErrorConstant_nonneg
lemma halasz_real_inner_approximation (f : ℕ → ℂ) (hf : OneBounded f)
(hmul : ∀ m n : ℕ, 0 < m → 0 < n → f (m * n) = f m * f n)
(M : ℕ) (hM : 1 ≤ M) {x : ℝ} (hx : 1 < x)
(hMx : (M : ℝ) ≤ x) (hxM : x ≤ 2 * M) :
‖(∑ n ∈ Icc 1 M, f n) - halaszPrimeConvolution f M / (Real.log x : ℂ)‖ ≤
halaszInnerErrorConstant * x / Real.log x := by
have hlog : 0 < Real.log x := Real.log_pos hx
have hc : (Real.log x : ℂ) ≠ 0 := by exact_mod_cast ne_of_gt hlog
have he : (∑ n ∈ Icc 1 M, f n) - halaszPrimeConvolution f M / (Real.log x : ℂ) =
((∑ n ∈ Icc 1 M, f n) * (Real.log x : ℂ) -
halaszPrimeConvolution f M) / (Real.log x : ℂ) := by field_simp
rw [he, norm_div, Complex.norm_real, Real.norm_eq_abs, abs_of_pos hlog]
exact div_le_div_of_nonneg_right (halasz_real_log_approximation f hf hmul M hM hMx hxM)
hlog.le
end TwoPointCorrelations
end OAI
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