| import OAI.NumberTheory.TwoPointCorrelations.HalaszDistanceCutoff | |
| namespace OAI | |
| /-! The elementary losses when changing the prime cutoff by a fixed power. -/ | |
| namespace TwoPointCorrelations | |
| lemma halasz_decay_shift {M K : ℝ} (hM : 0 ≤ M) (hK : 0 ≤ K) : | |
| (max 0 (M - K) + 1) * Real.exp (-max 0 (M - K)) ≤ | |
| Real.exp K * ((M + 1) * Real.exp (-M)) := by | |
| by_cases hMK : M ≤ K | |
| · rw [max_eq_left (by linarith : M - K ≤ 0)] | |
| simp only [zero_add, neg_zero, Real.exp_zero, mul_one] | |
| have he : 1 ≤ Real.exp K * Real.exp (-M) := by | |
| rw [← Real.exp_add] | |
| exact Real.one_le_exp_iff.mpr (by linarith) | |
| have hmul := mul_le_mul_of_nonneg_right (show 1 ≤ M + 1 by linarith) | |
| (mul_nonneg (Real.exp_nonneg K) (Real.exp_nonneg (-M))) | |
| nlinarith | |
| · rw [max_eq_right (by linarith : 0 ≤ M - K)] | |
| have he : Real.exp (-(M - K)) = Real.exp K * Real.exp (-M) := by | |
| rw [← Real.exp_add] | |
| congr 1 | |
| ring | |
| rw [he] | |
| have hh := mul_le_mul_of_nonneg_right (show M - K + 1 ≤ M + 1 by linarith) | |
| (mul_nonneg (Real.exp_nonneg K) (Real.exp_nonneg (-M))) | |
| nlinarith | |
| lemma halasz_log_error_cutoff {n N : ℕ} (hn : 2 ≤ n) (hnN : n ≤ N) | |
| (hNc : N ≤ n ^ 3) (hllN : 0 ≤ Real.log (Real.log (N : ℝ))) : | |
| Real.log (Real.log (n : ℝ)) / Real.log n ≤ | |
| 3 * (Real.log (Real.log (N : ℝ)) / Real.log N) := by | |
| have hn0 : (0 : ℝ) < n := by exact_mod_cast (by omega : 0 < n) | |
| have hN0 : (0 : ℝ) < N := by exact_mod_cast (by omega : 0 < N) | |
| have hln : 0 < Real.log (n : ℝ) := Real.log_pos (by exact_mod_cast (by omega : 1 < n)) | |
| have hlN : 0 < Real.log (N : ℝ) := Real.log_pos (by exact_mod_cast (by omega : 1 < N)) | |
| have hlnN : Real.log (n : ℝ) ≤ Real.log (N : ℝ) := | |
| Real.log_le_log hn0 (by exact_mod_cast hnN) | |
| have hln3 : Real.log (N : ℝ) ≤ 3 * Real.log (n : ℝ) := by | |
| have h := Real.log_le_log hN0 (show (N : ℝ) ≤ (n : ℝ) ^ 3 by exact_mod_cast hNc) | |
| simpa only [Real.log_pow, Nat.cast_ofNat] using h | |
| calc | |
| _ ≤ Real.log (Real.log (N : ℝ)) / Real.log n := | |
| div_le_div_of_nonneg_right (Real.log_le_log hln hlnN) hln.le | |
| _ ≤ 3 * (Real.log (Real.log (N : ℝ)) / Real.log N) := by | |
| rw [← mul_div_assoc] | |
| apply (div_le_div_iff₀ hln hlN).mpr | |
| nlinarith [mul_le_mul_of_nonneg_left hln3 hllN] | |
| end TwoPointCorrelations | |
| end OAI | |