openAI-math-dataset / lean /OAI /NumberTheory /TwoPoint /ShortIntervals /MRTCharacterSmallHeight.lean
| import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTCharacterGlobalTail | |
| import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTLiouvilleLSeries | |
| import Mathlib.Analysis.SpecialFunctions.Pow.Asymptotics | |
| namespace OAI | |
| /-! A concrete low-height range in the original Liouville prime tail. | |
| Only nonprincipal characters are covered here. -/ | |
| namespace TwoPointCorrelations | |
| open Filter Finset | |
| open scoped Classical | |
| lemma mrt_prime_power_cutoff_log_lower {a : ℝ} (ha : 0 < a) : | |
| ∀ᶠ X : ℕ in atTop, | |
| (1 / 2 : ℝ) * (Real.log (X : ℝ)) ^ a ≤ | |
| Real.log (mrtPrimePowerCutoff a X : ℝ) := by | |
| have hlog : Tendsto (fun X : ℕ => Real.log (X : ℝ)) atTop atTop := | |
| Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop | |
| have hp := ((tendsto_rpow_atTop ha).comp hlog).eventually | |
| (eventually_ge_atTop (max 2 (2 * Real.log 2))) | |
| filter_upwards [hp] with X hX | |
| let A := (Real.log (X : ℝ)) ^ a | |
| have hA : 2 ≤ A := (le_max_left _ _).trans hX | |
| have hAl : 2 * Real.log 2 ≤ A := (le_max_right _ _).trans hX | |
| have hE : 2 ≤ Real.exp A := by linarith [Real.add_one_le_exp A] | |
| have hf : Real.exp A / 2 ≤ (mrtPrimePowerCutoff a X : ℝ) := by | |
| have hh := Nat.lt_floor_add_one (Real.exp A) | |
| change Real.exp A / 2 ≤ (⌊Real.exp A⌋₊ : ℝ) | |
| linarith | |
| have hf0 : 0 < (mrtPrimePowerCutoff a X : ℝ) := | |
| (div_pos (Real.exp_pos A) (by norm_num)).trans_le hf | |
| have hh := Real.log_le_log (div_pos (Real.exp_pos A) (by norm_num)) hf | |
| rw [Real.log_div (Real.exp_pos A).ne' (by norm_num), Real.log_exp] at hh | |
| change (1 / 2 : ℝ) * A ≤ _ | |
| linarith | |
| lemma mrt_character_small_height_bound {q : ℕ} [NeZero q] {L t : ℝ} | |
| (hL : 1 ≤ L) (hq : (q : ℝ) ≤ L ^ (1 / 125 : ℝ)) | |
| (ht : |t| ≤ Real.exp (L ^ (1 / 3 : ℝ))) | |
| (hlog : Real.log L ≤ L ^ (1 / 3 : ℝ)) | |
| (hthree : Real.log 3 ≤ L ^ (1 / 3 : ℝ)) : | |
| mrtCharacterHeight q t ≤ 3 * L ^ (1 / 3 : ℝ) := by | |
| have hL0 : 0 < L := zero_lt_one.trans_le hL | |
| have hqp : 0 < (q : ℝ) := by exact_mod_cast NeZero.pos q | |
| have hlogq := Real.log_le_log hqp hq | |
| rw [Real.log_rpow hL0] at hlogq | |
| have hE : 1 ≤ Real.exp (L ^ (1 / 3 : ℝ)) := | |
| Real.one_le_exp (Real.rpow_nonneg hL0.le _) | |
| have he : |t| + 2 ≤ 3 * Real.exp (L ^ (1 / 3 : ℝ)) := by linarith | |
| have hl := Real.log_le_log (by positivity : 0 < |t| + 2) he | |
| rw [Real.log_mul (by norm_num) (Real.exp_pos _).ne', Real.log_exp] at hl | |
| rw [mrt_character_height_split] | |
| have hpow : 0 ≤ L ^ (1 / 3 : ℝ) := Real.rpow_nonneg hL0.le _ | |
| linarith | |
| lemma mrt_character_small_height_cost : ∀ᶠ L : ℝ in atTop, | |
| ∀ (q : ℕ) [NeZero q], (q : ℝ) ≤ L ^ (1 / 125 : ℝ) → | |
| ∀ t : ℝ, |t| ≤ Real.exp (L ^ (1 / 3 : ℝ)) → | |
| (q : ℝ) ^ 2 * mrtCharacterHeight q t ^ 2 ≤ (1 / 2 : ℝ) * L ^ (3 / 4 : ℝ) := by | |
| have hl := (isLittleO_log_rpow_atTop (show (0 : ℝ) < 1 / 3 by norm_num)).bound | |
| (show (0 : ℝ) < 1 by norm_num) | |
| have hp := (tendsto_rpow_atTop (show (0 : ℝ) < 101 / 1500 by norm_num)).eventually | |
| (eventually_ge_atTop (18 : ℝ)) | |
| have hthree := (tendsto_rpow_atTop (show (0 : ℝ) < 1 / 3 by norm_num)).eventually | |
| (eventually_ge_atTop (Real.log 3)) | |
| filter_upwards [eventually_ge_atTop (1 : ℝ), hl, hp, hthree] with L hL hl hp hthree | |
| have hL0 : 0 < L := zero_lt_one.trans_le hL | |
| rw [Real.norm_eq_abs, abs_of_nonneg (Real.log_nonneg hL), Real.norm_eq_abs, | |
| abs_of_pos (Real.rpow_pos_of_pos hL0 _), one_mul] at hl | |
| intro q _ hq t ht | |
| have hh := mrt_character_small_height_bound hL hq ht hl hthree | |
| have hH := (mrt_character_height_pos q t).le | |
| have hq0 : 0 ≤ (q : ℝ) := Nat.cast_nonneg q | |
| have hm := mul_le_mul (pow_le_pow_left₀ hq0 hq 2) (pow_le_pow_left₀ hH hh 2) | |
| (sq_nonneg _) (by positivity : 0 ≤ (L ^ (1 / 125 : ℝ)) ^ 2) | |
| have h₁ : (L ^ (1 / 125 : ℝ)) ^ 2 = L ^ (2 / 125 : ℝ) := by | |
| rw [← Real.rpow_natCast, ← Real.rpow_mul hL0.le] | |
| norm_num | |
| have h₂ : (L ^ (1 / 3 : ℝ)) ^ 2 = L ^ (2 / 3 : ℝ) := by | |
| rw [← Real.rpow_natCast, ← Real.rpow_mul hL0.le] | |
| norm_num | |
| have he : (L ^ (1 / 125 : ℝ)) ^ 2 * (3 * L ^ (1 / 3 : ℝ)) ^ 2 = | |
| 9 * L ^ (256 / 375 : ℝ) := by | |
| calc | |
| _ = 9 * (L ^ (2 / 125 : ℝ) * L ^ (2 / 3 : ℝ)) := by rw [mul_pow, h₁, h₂]; ring | |
| _ = _ := by rw [← Real.rpow_add hL0]; norm_num | |
| rw [he] at hm | |
| have hmul := mul_le_mul_of_nonneg_right hp (Real.rpow_nonneg hL0.le (256 / 375)) | |
| have hsum : L ^ (101 / 1500 : ℝ) * L ^ (256 / 375 : ℝ) = L ^ (3 / 4 : ℝ) := by | |
| rw [← Real.rpow_add hL0] | |
| norm_num | |
| rw [hsum] at hmul | |
| linarith | |
| theorem mrt_liouville_nonprincipal_small_height : ∃ C : ℝ, 0 < C ∧ | |
| ∀ᶠ X : ℕ in atTop, ∀ q : ℕ, 0 < q → | |
| (q : ℝ) ≤ (Real.log (X : ℝ)) ^ (1 / 125 : ℝ) → | |
| ∀ χ : DirichletCharacter ℂ q, χ ≠ 1 → ∀ t : ℝ, | |
| |t| ≤ Real.exp ((Real.log (X : ℝ)) ^ (1 / 3 : ℝ)) → | |
| |(∑ p ∈ mrtLiouvillePrimeTail X, characterTwist χ t p / (p : ℂ)).re| ≤ C := by | |
| obtain ⟨C, hC, htail⟩ := mrt_character_global_prime_tail | |
| refine ⟨C, hC, ?_⟩ | |
| have hlog : Tendsto (fun X : ℕ => Real.log (X : ℝ)) atTop atTop := | |
| Real.tendsto_log_atTop.comp tendsto_natCast_atTop_atTop | |
| filter_upwards [hlog.eventually mrt_character_small_height_cost, | |
| mrt_prime_power_cutoff_log_lower (show (0 : ℝ) < 3 / 4 by norm_num), | |
| mrt_prime_power_cutoff_eventually (show (0 : ℝ) < 3 / 4 by norm_num) (by norm_num)] | |
| with X hcost hlower hcut | |
| intro q hq hqX χ hχ t ht | |
| let : NeZero q := ⟨Nat.ne_of_gt hq⟩ | |
| change |(∑ p ∈ mrtPrimePowerTail (3 / 4) X, characterTwist χ t p / (p : ℂ)).re| ≤ C | |
| rw [mrt_prime_power_tail_cutoff] | |
| exact htail q χ hχ t _ _ hcut.1 hcut.2 ((hcost q hqX t ht).trans hlower) | |
| end TwoPointCorrelations | |
| end OAI | |