| import OAI.NumberTheory.TwoPointCorrelations.HalaszMeanValue |
| import OAI.NumberTheory.TwoPointCorrelations.HalaszCorrectionTransfer |
| import OAI.NumberTheory.TwoPointCorrelations.HalaszCorrectionScale |
| import OAI.NumberTheory.TwoPointCorrelations.HalaszDistanceDecay |
|
|
| namespace OAI |
|
|
| /-! The ordinary Halász estimate for general normalized multiplicative |
| functions. Only the values at primes enter the pretentious hypothesis. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset Filter |
|
|
| lemma halasz_complete_distance (f : ℕ → ℂ) (g : ℕ → ℂ) (N : ℕ) : |
| squaredDistance (mrtCompletePart f) g N = squaredDistance f g N := by |
| unfold squaredDistance |
| apply sum_congr rfl |
| intro p hp |
| have hprime : p.Prime := (mem_filter.mp hp).2 |
| have he : mrtCompletePart f p = f p := by |
| simpa using mrtCompletePart_prime_pow f hprime 1 |
| rw [he] |
|
|
| theorem halasz_general_mean_value : ∃ C X₀ : ℝ, 0 < C ∧ |
| ∀ (N : ℕ), X₀ ≤ N → |
| ∀ (f : ℕ → ℂ), f 1 = 1 → Multiplicative f → OneBounded f → |
| ∀ (M : ℝ), 0 ≤ M → |
| (∀ t ∈ Set.Icc (-(Real.log (N : ℝ) ^ 8)) (Real.log (N : ℝ) ^ 8), |
| M ≤ squaredDistance f (mrtArchimedeanTwist t) N) → |
| ‖∑ n ∈ Icc 1 N, f n‖ ≤ C * N * |
| ((M + 1) * Real.exp (-M) + Real.log (Real.log N) / Real.log N) := by |
| obtain ⟨C₀, Y₀, hC₀, hmean⟩ := halasz_mean_value |
| obtain ⟨K, hK, hcut⟩ := halasz_distance_cutoff_loss |
| let D := C₀ * Real.exp K + 3 * C₀ + 1 |
| let C := mrtCorrectionBound * D |
| have hCorr : 0 < mrtCorrectionBound := Real.exp_pos _ |
| have hD : 0 < D := by dsimp [D]; positivity |
| have hC : 0 < C := mul_pos hCorr hD |
| obtain ⟨X₀, hX₀⟩ := eventually_atTop.mp (halasz_eventually_correction_scale Y₀) |
| refine ⟨C, X₀, hC, ?_⟩ |
| intro N hXN f hf1 hf hb M hM hd |
| obtain ⟨hW, hWY, hWN, htail, hlog, hll⟩ := hX₀ N hXN |
| let W := ⌈Real.log (N : ℝ) ^ 4⌉₊ |
| let M' := max 0 (M - K) |
| let E := (M + 1) * Real.exp (-M) |
| let R := Real.log (Real.log (N : ℝ)) / Real.log N |
| let A := C₀ * (Real.exp K * E + 3 * R) |
| have hE : 0 ≤ E := by dsimp [E]; positivity |
| have hR : 0 ≤ R := by dsimp [R]; positivity |
| have hA : 0 ≤ A := by dsimp [A]; positivity |
| have hNlog : 0 < Real.log (N : ℝ) := by linarith |
| have hprefix (d : ℕ) (hdN : d ∈ Icc 1 N) (hdW : d ≤ W) : |
| ‖∑ n ∈ Icc 1 (N / d), mrtCompletePart f n‖ ≤ A * ((N / d : ℕ) : ℝ) := by |
| have hd0 : 0 < d := (mem_Icc.mp hdN).1 |
| have hWNnat : W ^ 2 ≤ N := by exact_mod_cast hWN |
| obtain ⟨hWq, hqN, hNq⟩ := halasz_correction_quotient hW hd0 hdW hWNnat |
| have hq2 : 2 ≤ N / d := hW.trans hWq |
| have hqY : Y₀ ≤ ((N / d : ℕ) : ℝ) := |
| hWY.trans (by exact_mod_cast hWq) |
| have hlnq : 0 < Real.log ((N / d : ℕ) : ℝ) := |
| Real.log_pos (by exact_mod_cast (show 1 < N / d by omega)) |
| have hlogqN : Real.log ((N / d : ℕ) : ℝ) ≤ Real.log (N : ℝ) := |
| Real.log_le_log (by exact_mod_cast (show 0 < N / d by omega)) (by exact_mod_cast hqN) |
| have hheight : Real.log ((N / d : ℕ) : ℝ) ^ 8 ≤ Real.log (N : ℝ) ^ 8 := |
| pow_le_pow_left₀ hlnq.le hlogqN 8 |
| have hdist (t : ℝ) (ht : t ∈ Set.Icc |
| (-(Real.log ((N / d : ℕ) : ℝ) ^ 8)) (Real.log ((N / d : ℕ) : ℝ) ^ 8)) : |
| M' ≤ squaredDistance (mrtCompletePart f) (mrtArchimedeanTwist t) (N / d) := by |
| rw [halasz_complete_distance] |
| apply max_le |
| · exact halasz_distance_nonneg f hb _ t |
| · have hl := hcut f hb (N / d) N hq2 hqN hNq t |
| have hu := hd t ⟨by linarith [ht.1], ht.2.trans hheight⟩ |
| linarith |
| have hm := hmean (N / d) hqY (mrtCompletePart f) (fromPrimePowers_one _) |
| (fun a b ha hb => mrtCompletePart_mul f ha hb) (mrtCompletePart_oneBounded f hb) |
| M' (le_max_left _ _) hdist |
| have hdec := halasz_decay_shift hM hK |
| have herr := halasz_log_error_cutoff hq2 hqN hNq (by linarith) |
| calc |
| _ ≤ C₀ * ((N / d : ℕ) : ℝ) * |
| ((M' + 1) * Real.exp (-M') + |
| Real.log (Real.log ((N / d : ℕ) : ℝ)) / Real.log ((N / d : ℕ) : ℝ)) := hm |
| _ ≤ C₀ * ((N / d : ℕ) : ℝ) * (Real.exp K * E + 3 * R) := by gcongr |
| _ = _ := by ring |
| have hc := halasz_correction_mean_bound f hf hb hf1 N W (by omega) A hA hprefix |
| have hRtail : (W : ℝ) ^ (-(1 / 4 : ℝ)) ≤ R := by |
| apply htail.trans |
| exact div_le_div_of_nonneg_right hll hNlog.le |
| have hsmall : A + (W : ℝ) ^ (-(1 / 4 : ℝ)) ≤ D * (E + R) := by |
| dsimp [A, D] |
| have hCE : 0 ≤ C₀ * Real.exp K * R := by positivity |
| have h3E : 0 ≤ 3 * C₀ * E := by positivity |
| nlinarith |
| calc |
| _ ≤ mrtCorrectionBound * N * (A + (W : ℝ) ^ (-(1 / 4 : ℝ))) := hc |
| _ ≤ mrtCorrectionBound * N * (D * (E + R)) := |
| mul_le_mul_of_nonneg_left hsmall (by positivity) |
| _ = _ := by dsimp [C, E, R]; ring |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
