| import OAI.NumberTheory.TwoPointCorrelations.HalaszBandSummation |
| import Mathlib.Analysis.SpecialFunctions.Log.Basic |
|
|
| namespace OAI |
|
|
| /-! The logarithmic truncation height and integer smoothing length used |
| in the grouped Perron estimate. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Filter |
|
|
| lemma halasz_eventually_numeric_scale (B : ℝ) : |
| ∀ᶠ x : ℝ in atTop, 3 ≤ x ∧ 1 ≤ Real.log (Real.log x) ∧ |
| 2 ≤ Real.log x ^ 4 ∧ Real.log x ^ 16 ≤ x / 2 ∧ B ≤ Real.log x ^ 8 := by |
| have hsmall := (Real.isLittleO_pow_log_id_atTop (n := 16)).bound |
| (by norm_num : (0 : ℝ) < 1 / 2) |
| have hlog := Real.tendsto_log_atTop.eventually (eventually_ge_atTop (max 2 B)) |
| have hloglog := (Real.tendsto_log_atTop.comp Real.tendsto_log_atTop).eventually |
| (eventually_ge_atTop (1 : ℝ)) |
| filter_upwards [eventually_ge_atTop (3 : ℝ), hlog, hloglog, hsmall] with x hx hl hll hs |
| have hl2 : 2 ≤ Real.log x := (le_max_left _ _).trans hl |
| have hl1 : 1 ≤ Real.log x := by linarith |
| dsimp only [id] at hs |
| rw [Real.norm_eq_abs, abs_of_nonneg (pow_nonneg (by linarith : 0 ≤ Real.log x) 16), |
| Real.norm_eq_abs, abs_of_pos (by linarith : 0 < x)] at hs |
| refine ⟨hx, hll, ?_, by linarith, ?_⟩ |
| · exact hl2.trans (le_self_pow₀ hl1 (by norm_num : (4 : ℕ) ≠ 0)) |
| · exact ((le_max_right _ _).trans hl).trans (le_self_pow₀ hl1 (by norm_num : (8 : ℕ) ≠ 0)) |
|
|
| lemma halasz_smoothing_integer {N : ℕ} (hN : 2 ≤ N) |
| (hlog : 2 ≤ Real.log (N : ℝ) ^ 4) (hsmall : Real.log (N : ℝ) ^ 4 ≤ N) : |
| let m := ⌈(N : ℝ) / Real.log (N : ℝ) ^ 4⌉₊ |
| 0 < m ∧ m ≤ N ∧ (N : ℝ) / Real.log (N : ℝ) ^ 4 ≤ m ∧ |
| (m : ℝ) ≤ 2 * ((N : ℝ) / Real.log (N : ℝ) ^ 4) ∧ |
| ((N : ℝ) + 1 / 2) / m ≤ 2 * Real.log (N : ℝ) ^ 4 := by |
| let L := Real.log (N : ℝ) ^ 4 |
| let m := ⌈(N : ℝ) / L⌉₊ |
| have hL : 0 < L := by dsimp [L]; linarith |
| have hNr : (2 : ℝ) ≤ N := by exact_mod_cast hN |
| have hratio : 1 ≤ (N : ℝ) / L := (le_div_iff₀ hL).mpr (by simpa [L] using hsmall) |
| have hratioHigh : (N : ℝ) / L ≤ (N : ℝ) / 2 := |
| div_le_div_of_nonneg_left (by positivity) (by norm_num) hlog |
| have hmlo : (N : ℝ) / L ≤ m := Nat.le_ceil _ |
| have hmhi : (m : ℝ) < (N : ℝ) / L + 1 := Nat.ceil_lt_add_one (by positivity) |
| have hmp : (0 : ℝ) < m := by linarith |
| have hmNat : 0 < m := by exact_mod_cast hmp |
| have hmN : m ≤ N := by exact_mod_cast (show (m : ℝ) ≤ N by linarith) |
| refine ⟨hmNat, hmN, hmlo, by linarith, ?_⟩ |
| apply (div_le_iff₀ hmp).mpr |
| have hmul : (N : ℝ) ≤ (m : ℝ) * L := (div_le_iff₀ hL).mp hmlo |
| dsimp [L] at hmul ⊢ |
| nlinarith |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
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|
