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180bc03 | 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 25 26 27 28 29 30 31 32 33 34 35 36 37 38 39 40 41 42 43 44 45 46 47 48 49 50 51 52 53 54 55 | 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
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