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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 56 57 58 59 60 61 62 63 64 65 66 67 68 69 70 71 72 73 74 75 76 77 78 79 80 81 82 83 84 85 86 87 88 89 90 91 92 93 94 95 96 97 98 99 100 101 102 103 104 105 106 107 108 109 | import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTCharacterHighZeros
import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTCharacterZeroCount
namespace OAI
/-! Horizontal separation of all zeros in the normalized disk from a
point in a narrower high-height strip. -/
namespace TwoPointCorrelations
open Complex
open scoped Classical
variable {q : ℕ} [NeZero q]
lemma mrt_character_physical_zero_height (χ : DirichletCharacter ℂ q) (t : ℝ)
{rho : ℂ} (hrho : rho ∈ mrtCharacterNormalizedZeros χ t) :
|(mrtCharacterPhysicalPoint t rho).im| ≤ |t| + 2 := by
have hn : ‖rho‖ ≤ 7 / 8 := by simpa using hrho.1
have hi := (Complex.abs_im_le_norm rho).trans hn
have he : (mrtCharacterPhysicalPoint t rho).im = t + (3 / 2 : ℝ) * rho.im := by
simp [mrtCharacterPhysicalPoint, Complex.mul_im]
rw [he]
calc
_ ≤ |t| + |(3 / 2 : ℝ) * rho.im| := abs_add_le _ _
_ = |t| + (3 / 2 : ℝ) * |rho.im| := by rw [abs_mul]; norm_num
_ ≤ _ := by linarith
lemma mrt_character_physical_zero_height_lower (χ : DirichletCharacter ℂ q) (t : ℝ)
{rho : ℂ} (hrho : rho ∈ mrtCharacterNormalizedZeros χ t) :
|t| ≤ |(mrtCharacterPhysicalPoint t rho).im| + 2 := by
have hn : ‖rho‖ ≤ 7 / 8 := by simpa using hrho.1
have hi := (Complex.abs_im_le_norm rho).trans hn
have he : t = (mrtCharacterPhysicalPoint t rho).im - (3 / 2 : ℝ) * rho.im := by
simp [mrtCharacterPhysicalPoint, Complex.mul_im]
calc
_ = |(mrtCharacterPhysicalPoint t rho).im - (3 / 2 : ℝ) * rho.im| := congrArg abs he
_ ≤ |(mrtCharacterPhysicalPoint t rho).im| + |(3 / 2 : ℝ) * rho.im| := abs_sub _ _
_ = |(mrtCharacterPhysicalPoint t rho).im| + (3 / 2 : ℝ) * |rho.im| := by
rw [abs_mul]; norm_num
_ ≤ _ := by linarith
lemma mrt_character_physical_zero_log_height (χ : DirichletCharacter ℂ q) (t : ℝ)
{rho : ℂ} (hrho : rho ∈ mrtCharacterNormalizedZeros χ t) :
mrtCharacterHeight q (mrtCharacterPhysicalPoint t rho).im ≤
2 * mrtCharacterHeight q t := by
have hh : Real.log (|(mrtCharacterPhysicalPoint t rho).im| + 2) ≤
2 * Real.log (|t| + 2) := by
calc
_ ≤ Real.log ((|t| + 2) ^ 2) := by
apply Real.log_le_log (by positivity)
have h := mrt_character_physical_zero_height χ t hrho
nlinarith [abs_nonneg t, sq_nonneg (|t|)]
_ = _ := by rw [Real.log_pow]; norm_num
have hq : 0 ≤ Real.log (q : ℝ) :=
Real.log_nonneg (by exact_mod_cast NeZero.pos q)
rw [mrt_character_height_split, mrt_character_height_split]
linarith
lemma mrt_character_zero_distance_of_strip {c T : ℝ} (hc : 0 < c)
(χ : DirichletCharacter ℂ q)
(hfree : ∀ t beta : ℝ, T ≤ |t| →
1 - c / mrtCharacterHeight q t ≤ beta →
DirichletCharacter.LFunction χ ((beta : ℂ) + Complex.I * (t : ℂ)) ≠ 0)
(t sigma : ℝ) (ht : T + 2 ≤ |t|)
(hsigma : 1 - c / (4 * mrtCharacterHeight q t) ≤ sigma)
{rho : ℂ} (hrho : rho ∈ mrtCharacterNormalizedZeros χ t) :
c / (6 * mrtCharacterHeight q t) ≤ ‖mrtCharacterRealDiskPoint sigma - rho‖ := by
let v := mrtCharacterPhysicalPoint t rho
have hzero : DirichletCharacter.LFunction χ ((v.re : ℂ) + Complex.I * (v.im : ℂ)) = 0 := by
have he : (v.re : ℂ) + Complex.I * (v.im : ℂ) = v := by
apply Complex.ext <;> simp
rw [he]
exact mrtCharacterNormalizedZeros_physical_zero χ t hrho
have hvt : T ≤ |v.im| := by
have hh := mrt_character_physical_zero_height_lower χ t hrho
change |t| ≤ |v.im| + 2 at hh
linarith
have hv : v.re < 1 - c / mrtCharacterHeight q v.im := by
by_contra! h
exact hfree v.im v.re hvt h hzero
have hH := mrt_character_height_pos q t
have hvH := mrt_character_height_pos q v.im
have hlog := mrt_character_physical_zero_log_height χ t hrho
have hinv : c / (2 * mrtCharacterHeight q t) ≤ c / mrtCharacterHeight q v.im :=
div_le_div_of_nonneg_left hc.le hvH hlog
have hgap : c / (4 * mrtCharacterHeight q t) ≤ sigma - v.re := by
have he : c / (2 * mrtCharacterHeight q t) =
2 * (c / (4 * mrtCharacterHeight q t)) := by ring
rw [he] at hinv
linarith
have hre : (mrtCharacterRealDiskPoint sigma - rho).re =
(2 / 3 : ℝ) * (sigma - v.re) := by
dsimp [mrtCharacterRealDiskPoint, v, mrtCharacterPhysicalPoint]
simp only [Complex.ofReal_re, Complex.mul_re,
Complex.I_re, Complex.I_im, Complex.ofReal_im]
norm_num
ring
calc
_ = (2 / 3 : ℝ) * (c / (4 * mrtCharacterHeight q t)) := by ring
_ ≤ (2 / 3 : ℝ) * (sigma - v.re) := mul_le_mul_of_nonneg_left hgap (by norm_num)
_ = (mrtCharacterRealDiskPoint sigma - rho).re := hre.symm
_ ≤ |(mrtCharacterRealDiskPoint sigma - rho).re| := le_abs_self _
_ ≤ _ := Complex.abs_re_le_norm _
end TwoPointCorrelations
end OAI
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