| import OAI.NumberTheory.TwoPointCorrelations.HalaszPrimeConvolution |
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| namespace OAI |
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| /-! A complex-valued approximation, not just a norm bound, permits a second |
| application of the prime convolution inside the first one. -/ |
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| namespace TwoPointCorrelations |
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| open Finset |
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| 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 |
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| 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) |
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| noncomputable def halaszInnerErrorConstant : ℝ := |
| halaszPrimePowerConstant + 1 + Real.log 2 |
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| 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 |
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|
| / |
| 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 |
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| 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 |
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| end TwoPointCorrelations |
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| end OAI |
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