openAI-math-dataset / lean /OAI /NumberTheory /TwoPoint /ShortIntervals /MRTCharacterZeroDistance.lean
| 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 | |