openAI-math-dataset / lean /OAI /NumberTheory /TwoPoint /ShortIntervals /MRTCharacterMovingCenter.lean
| import OAI.NumberTheory.TwoPoint.ShortIntervals.MRTAdaptiveDisk | |
| namespace OAI | |
| /-! Normalization of Dirichlet L-functions on a moving disk. The radius is | |
| an explicit parameter; all characters, including the principal character, | |
| are allowed when the disk stays away from the real axis. -/ | |
| namespace TwoPointCorrelations | |
| open Complex Filter | |
| open scoped BigOperators Classical Topology | |
| variable {q : ℕ} [NeZero q] | |
| noncomputable def mrtMovingPoint (r t : ℝ) (z : ℂ) : ℂ := | |
| ((1 + 2 * r : ℝ) : ℂ) + Complex.I * (t : ℂ) + ((3 * r : ℝ) : ℂ) * z | |
| noncomputable def mrtMovingLFunction (χ : DirichletCharacter ℂ q) (r t : ℝ) | |
| (z : ℂ) : ℂ := | |
| DirichletCharacter.LFunction χ (mrtMovingPoint r t z) / | |
| DirichletCharacter.LFunction χ (mrtMovingPoint r t 0) | |
| lemma mrt_moving_point_ne_one {r t : ℝ} (hr : 0 < r) (ht : 3 * r < |t|) | |
| {z : ℂ} (hz : ‖z‖ ≤ 1) : mrtMovingPoint r t z ≠ 1 := by | |
| intro he | |
| have him := congrArg Complex.im he | |
| have him' : t + 3 * r * z.im = 0 := by | |
| simpa [mrtMovingPoint, Complex.mul_im] using him | |
| have hzi := (Complex.abs_im_le_norm z).trans hz | |
| have heq : t = -(3 * r * z.im) := by linarith | |
| rw [heq, abs_neg, abs_mul, abs_of_pos (by positivity : 0 < 3 * r)] at ht | |
| nlinarith | |
| lemma mrt_moving_point_center_ne_zero (χ : DirichletCharacter ℂ q) | |
| {r : ℝ} (hr : 0 < r) (t : ℝ) : | |
| DirichletCharacter.LFunction χ (mrtMovingPoint r t 0) ≠ 0 := by | |
| apply χ.LFunction_ne_zero_of_one_le_re | |
| · right | |
| intro h | |
| have hh := congrArg Complex.re h | |
| simp [mrtMovingPoint] at hh | |
| linarith | |
| · simp [mrtMovingPoint] | |
| linarith | |
| lemma mrt_moving_LFunction_zero (χ : DirichletCharacter ℂ q) | |
| {r : ℝ} (hr : 0 < r) (t : ℝ) : mrtMovingLFunction χ r t 0 = 1 := | |
| div_self (mrt_moving_point_center_ne_zero χ hr t) | |
| lemma mrt_moving_LFunction_analytic (χ : DirichletCharacter ℂ q) | |
| {r t : ℝ} (hr : 0 < r) (ht : 3 * r < |t|) : | |
| ∀ z ∈ Metric.closedBall (0 : ℂ) 1, AnalyticAt ℂ (mrtMovingLFunction χ r t) z := by | |
| intro z hz | |
| have hn := mrt_moving_point_ne_one hr ht (by simpa using hz) | |
| have hd : DifferentiableOn ℂ (DirichletCharacter.LFunction χ) ({1}ᶜ : Set ℂ) := by | |
| intro w hw | |
| exact (χ.differentiableAt_LFunction w (Or.inl hw)).differentiableWithinAt | |
| have ha := hd.analyticAt (isOpen_compl_singleton.mem_nhds hn) | |
| have hp : AnalyticAt ℂ (mrtMovingPoint r t) z := by | |
| apply Differentiable.analyticAt | |
| unfold mrtMovingPoint | |
| fun_prop | |
| exact (ha.comp hp).div_const | |
| lemma mrt_moving_LFunction_growth (χ : DirichletCharacter ℂ q) | |
| {r t A V B : ℝ} | |
| (hA : 0 ≤ A) | |
| (hg : ∀ z ∈ Metric.closedBall (0 : ℂ) (15 / 16), | |
| ‖DirichletCharacter.LFunction χ (mrtMovingPoint r t z)‖ ≤ A) | |
| (hi : ‖(DirichletCharacter.LFunction χ (mrtMovingPoint r t 0))⁻¹‖ ≤ V) | |
| (hB : A * V ≤ Real.exp B) : | |
| ∀ z ∈ Metric.closedBall (0 : ℂ) (15 / 16), | |
| ‖mrtMovingLFunction χ r t z‖ ≤ Real.exp B := by | |
| intro z hz | |
| unfold mrtMovingLFunction | |
| rw [div_eq_mul_inv, norm_mul] | |
| exact (mul_le_mul (hg z hz) hi (norm_nonneg _) hA).trans hB | |
| lemma mrt_moving_LFunction_deriv (χ : DirichletCharacter ℂ q) | |
| {r t : ℝ} (hr : 0 < r) (ht : 3 * r < |t|) | |
| {z : ℂ} (hz : ‖z‖ ≤ 1) : | |
| deriv (mrtMovingLFunction χ r t) z = | |
| (deriv (DirichletCharacter.LFunction χ) (mrtMovingPoint r t z) * | |
| ((3 * r : ℝ) : ℂ)) / | |
| DirichletCharacter.LFunction χ (mrtMovingPoint r t 0) := by | |
| have hp : HasDerivAt (mrtMovingPoint r t) ((3 * r : ℝ) : ℂ) z := by | |
| exact (hasDerivAt_const_mul ((3 * r : ℝ) : ℂ)).const_add | |
| (((1 + 2 * r : ℝ) : ℂ) + Complex.I * (t : ℂ)) | |
| have hn := mrt_moving_point_ne_one hr ht hz | |
| exact (((χ.differentiableAt_LFunction _ (Or.inl hn)).hasDerivAt.comp z hp).div_const | |
| (DirichletCharacter.LFunction χ (mrtMovingPoint r t 0))).deriv | |
| lemma mrt_moving_logderiv_eq (χ : DirichletCharacter ℂ q) | |
| {r t : ℝ} (hr : 0 < r) (ht : 3 * r < |t|) | |
| {z : ℂ} (hz : ‖z‖ ≤ 1) | |
| (hn : DirichletCharacter.LFunction χ (mrtMovingPoint r t z) ≠ 0) : | |
| deriv (mrtMovingLFunction χ r t) z / mrtMovingLFunction χ r t z = | |
| ((3 * r : ℝ) : ℂ) * | |
| (deriv (DirichletCharacter.LFunction χ) (mrtMovingPoint r t z) / | |
| DirichletCharacter.LFunction χ (mrtMovingPoint r t z)) := by | |
| rw [mrt_moving_LFunction_deriv χ hr ht hz] | |
| unfold mrtMovingLFunction | |
| field_simp [hn, mrt_moving_point_center_ne_zero χ hr t] | |
| /-- A separated normalized zero set bounds the actual logarithmic derivative. | |
| The factor `1/(3r)` is the only cost of changing the disk radius. -/ | |
| theorem mrt_moving_logderiv_norm (χ : DirichletCharacter ℂ q) | |
| {r t B η : ℝ} (hr : 0 < r) (ht : 3 * r < |t|) (hB : 0 < B) | |
| (hg : ∀ z ∈ Metric.closedBall (0 : ℂ) (15 / 16), | |
| ‖mrtMovingLFunction χ r t z‖ ≤ Real.exp B) | |
| {z : ℂ} (hz : ‖z‖ ≤ 3 / 4) | |
| (hn : DirichletCharacter.LFunction χ (mrtMovingPoint r t z) ≠ 0) | |
| (hη : 0 < η) | |
| (hd : ∀ ρ ∈ mrtDiskZeros (mrtMovingLFunction χ r t), η ≤ ‖z - ρ‖) : | |
| ‖deriv (DirichletCharacter.LFunction χ) (mrtMovingPoint r t z) / | |
| DirichletCharacter.LFunction χ (mrtMovingPoint r t z)‖ ≤ | |
| ((mrtCharacterLogDerivativeConstant + | |
| (1 / Real.log ((15 / 16 : ℝ) / (7 / 8))) / η) * B) / (3 * r) := by | |
| have hf := mrt_moving_LFunction_analytic χ hr ht | |
| have h0 := mrt_moving_LFunction_zero χ hr t | |
| have hfn : mrtMovingLFunction χ r t z ≠ 0 := | |
| div_ne_zero hn (mrt_moving_point_center_ne_zero χ hr t) | |
| have hh := mrt_disk_logderiv_norm _ hf h0 hB hg hz hfn hη hd | |
| rw [mrt_moving_logderiv_eq χ hr ht (hz.trans (by norm_num)) hn, norm_mul, | |
| Complex.norm_real, Real.norm_eq_abs, abs_of_pos (by positivity : 0 < 3 * r)] at hh | |
| exact (le_div_iff₀ (by positivity : 0 < 3 * r)).mpr (by nlinarith only [hh]) | |
| end TwoPointCorrelations | |
| end OAI | |