| import OAI.NumberTheory.TwoPointCorrelations.HalaszLogBands |
| import Mathlib.Data.Nat.Log |
|
|
| namespace OAI |
|
|
| /-! A finite geometric partition of log(X/p), using base-two bands. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| noncomputable def halaszBandIndex (y : ℝ) : ℕ := Nat.log 2 ⌊y / Real.log 2⌋₊ |
|
|
| noncomputable def halaszBandScale (j : ℕ) : ℝ := Real.log 2 * 2 ^ j |
|
|
| lemma halasz_band_scale_bounds {y : ℝ} (hy : Real.log 2 ≤ y) : |
| halaszBandScale (halaszBandIndex y) ≤ y ∧ |
| y < 2 * halaszBandScale (halaszBandIndex y) := by |
| have htwo : 0 < Real.log 2 := Real.log_pos (by norm_num) |
| have hy0 : 0 ≤ y / Real.log 2 := div_nonneg (htwo.le.trans hy) htwo.le |
| have hfloor : 1 ≤ ⌊y / Real.log 2⌋₊ := |
| (Nat.le_floor_iff hy0).mpr ((le_div_iff₀ htwo).mpr (by simpa using hy)) |
| have hlow := Nat.pow_log_le_self 2 (by omega : ⌊y / Real.log 2⌋₊ ≠ 0) |
| have hhigh := Nat.lt_pow_succ_log_self (by norm_num : 1 < 2) ⌊y / Real.log 2⌋₊ |
| have hlowR : (2 : ℝ) ^ halaszBandIndex y ≤ (⌊y / Real.log 2⌋₊ : ℝ) := by |
| exact_mod_cast hlow |
| have hhighR : (⌊y / Real.log 2⌋₊ : ℝ) + 1 ≤ (2 : ℝ) ^ (halaszBandIndex y + 1) := by |
| exact_mod_cast hhigh |
| constructor |
| · have h := hlowR.trans (Nat.floor_le hy0) |
| have hh := (le_div_iff₀ htwo).mp h |
| simpa only [halaszBandScale, mul_comm] using hh |
| · have h := (Nat.lt_floor_add_one (y / Real.log 2)).trans_le hhighR |
| have hh := (div_lt_iff₀ htwo).mp h |
| simpa only [halaszBandScale, pow_succ, mul_assoc, mul_comm, mul_left_comm] using hh |
|
|
| lemma halasz_band_index_mono {y Y : ℝ} (hyY : y ≤ Y) : |
| halaszBandIndex y ≤ halaszBandIndex Y := by |
| apply Nat.log_mono_right |
| apply Nat.floor_mono |
| exact div_le_div_of_nonneg_right hyY (le_of_lt (Real.log_pos (by norm_num : (1 : ℝ) < 2))) |
|
|
| noncomputable def halaszPrimeBandPart (P : Finset ℕ) (X : ℝ) (j : ℕ) : Finset ℕ := |
| P.filter (fun p : ℕ => halaszBandIndex (Real.log (X / p)) = j) |
|
|
| lemma halasz_prime_band_part_bounds (P : Finset ℕ) {X : ℝ} {j p : ℕ} |
| (hP : ∀ p ∈ P, Real.log 2 ≤ Real.log (X / p)) |
| (hp : p ∈ halaszPrimeBandPart P X j) : |
| halaszBandScale j ≤ Real.log (X / p) ∧ Real.log (X / p) < 2 * halaszBandScale j := by |
| obtain ⟨hpP, hpj⟩ := mem_filter.mp hp |
| simpa only [hpj] using halasz_band_scale_bounds (hP p hpP) |
|
|
| theorem halasz_prime_band_partition (P : Finset ℕ) {X Y : ℝ} |
| (hP : ∀ p ∈ P, Real.log (X / p) ≤ Y) (a : ℕ → ℂ) : |
| (∑ p ∈ P, a p) = |
| ∑ j ∈ range (halaszBandIndex Y + 1), ∑ p ∈ halaszPrimeBandPart P X j, a p := by |
| symm |
| apply sum_fiberwise_of_maps_to |
| intro p hp |
| exact mem_range.mpr (Nat.lt_succ_of_le (halasz_band_index_mono (hP p hp))) |
|
|
| lemma halasz_band_scale_ge_log_two (j : ℕ) : Real.log 2 ≤ halaszBandScale j := by |
| have hp : (1 : ℝ) ≤ 2 ^ j := one_le_pow₀ (by norm_num) |
| simpa only [mul_one, halaszBandScale] using |
| mul_le_mul_of_nonneg_left hp (le_of_lt (Real.log_pos (by norm_num : (1 : ℝ) < 2))) |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
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|
