| import Mathlib.Analysis.Complex.JensenFormula |
| import OAI.NumberTheory.Ostmann.Characters.CharacterMellinBound |
|
|
| namespace OAI |
|
|
| /-! # Local zero counts for primitive character L-functions |
|
|
| Jensen's formula is applied to disks centered on Re(s)=2. The larger |
| radius stays in the positive half-plane; the smaller disk covers a unit |
| ordinate interval in the right half of the critical strip. |
| -/ |
|
|
| namespace Ostmann |
|
|
| open Metric Set MeromorphicOn |
| open scoped BigOperators |
|
|
| noncomputable def characterZeroCenter (t : ℝ) : ℂ := 2 + (t : ℂ) * Complex.I |
|
|
| @[simp] theorem characterZeroCenter_re (t : ℝ) : (characterZeroCenter t).re = 2 := by |
| simp [characterZeroCenter] |
|
|
| @[simp] theorem characterZeroCenter_im (t : ℝ) : (characterZeroCenter t).im = t := by |
| simp [characterZeroCenter] |
|
|
| theorem characterZeroCenter_norm (t : ℝ) : ‖characterZeroCenter t‖ ≤ 2 + |t| := by |
| simpa using Complex.norm_le_abs_re_add_abs_im (characterZeroCenter t) |
|
|
| theorem character_disk_re (t : ℝ) (z : ℂ) |
| (hz : z ∈ closedBall (characterZeroCenter t) (7 / 4 : ℝ)) : 1 / 4 ≤ z.re := by |
| rw [mem_closedBall, dist_eq_norm] at hz |
| have hr := (Complex.abs_re_le_norm (z - characterZeroCenter t)).trans hz |
| simp only [Complex.sub_re, characterZeroCenter_re] at hr |
| linarith [(abs_le.mp hr).1] |
|
|
| theorem character_disk_norm (t : ℝ) (z : ℂ) |
| (hz : z ∈ closedBall (characterZeroCenter t) (7 / 4 : ℝ)) : ‖z‖ ≤ |t| + 4 := by |
| rw [mem_closedBall, dist_eq_norm] at hz |
| have htri := norm_add_le (z - characterZeroCenter t) (characterZeroCenter t) |
| rw [sub_add_cancel] at htri |
| linarith [characterZeroCenter_norm t] |
|
|
| theorem PrimitiveComplexCharacter.L_disk_bound (χ : PrimitiveComplexCharacter) |
| (t : ℝ) (z : ℂ) (hz : z ∈ closedBall (characterZeroCenter t) (7 / 4 : ℝ)) : |
| ‖χ.L z‖ ≤ 16 * (χ.modulus : ℝ) * (|t| + 2) := by |
| have hr := character_disk_re t z hz |
| have hnorm := character_disk_norm t z hz |
| have hq : (1 : ℝ) ≤ χ.modulus := by exact_mod_cast χ.positive |
| calc |
| _ ≤ ‖z‖ * ((χ.modulus : ℝ) + 1) / z.re := χ.L_norm_bound_positive z (by linarith) |
| _ ≤ ‖z‖ * ((χ.modulus : ℝ) + 1) / (1 / 4) := |
| div_le_div_of_nonneg_left (by positivity) (by norm_num) hr |
| _ ≤ (|t| + 4) * ((χ.modulus : ℝ) + 1) / (1 / 4) := by gcongr |
| _ ≤ _ := by |
| norm_num only [div_eq_mul_inv, inv_div, inv_one] |
| nlinarith [abs_nonneg t, mul_nonneg (sub_nonneg.mpr hq) (abs_nonneg t)] |
|
|
| noncomputable def characterLocalZeroMass (χ : PrimitiveComplexCharacter) (t : ℝ) : ℤ := |
| ∑ᶠ z, divisor χ.L (closedBall (characterZeroCenter t) (13 / 8 : ℝ)) z |
|
|
| / |
| theorem PrimitiveComplexCharacter.local_zero_mass_bound (χ : PrimitiveComplexCharacter) |
| (t : ℝ) : |
| (characterLocalZeroMass χ t : ℝ) ≤ |
| Real.log (48 * (χ.modulus : ℝ) * (|t| + 2)) / Real.log (14 / 13) := by |
| let : NeZero χ.modulus := ⟨χ.positive.ne'⟩ |
| have hq : (1 : ℝ) ≤ χ.modulus := by exact_mod_cast χ.positive |
| have ha : AnalyticOnNhd ℂ χ.L (closedBall (characterZeroCenter t) |(7 / 4 : ℝ)|) := |
| fun _ _ => (DirichletCharacter.differentiable_LFunction χ.nontrivial).analyticAt _ |
| have hlo := χ.L_lower_re_two (characterZeroCenter t) (characterZeroCenter_re t) |
| have hne : χ.L (characterZeroCenter t) ≠ 0 := by |
| intro he |
| rw [he, norm_zero] at hlo |
| linarith |
| have hM : 1 ≤ 16 * (χ.modulus : ℝ) * (|t| + 2) := by |
| nlinarith [abs_nonneg t, mul_nonneg (sub_nonneg.mpr hq) (abs_nonneg t)] |
| have hj := ha.sum_divisor_le (r := (13 / 8 : ℝ)) (R := (7 / 4 : ℝ)) |
| (by norm_num) (by norm_num) hM hne (fun z hz => |
| χ.L_disk_bound t z (by simpa only [abs_of_pos (by norm_num : (0 : ℝ) < 7 / 4)] using |
| sphere_subset_closedBall hz)) |
| have hratio : (16 * (χ.modulus : ℝ) * (|t| + 2)) / ‖χ.L (characterZeroCenter t)‖ ≤ |
| 48 * (χ.modulus : ℝ) * (|t| + 2) := by |
| apply (div_le_iff₀ (by linarith : 0 < ‖χ.L (characterZeroCenter t)‖)).2 |
| nlinarith [mul_le_mul_of_nonneg_left hlo |
| (show 0 ≤ 48 * (χ.modulus : ℝ) * (|t| + 2) by positivity)] |
| have hlog : 0 < Real.log (14 / 13 : ℝ) := Real.log_pos (by norm_num) |
| have hlogratio := Real.log_le_log (div_pos (by positivity) |
| (by linarith : 0 < ‖χ.L (characterZeroCenter t)‖)) hratio |
| rw [show |(13 / 8 : ℝ)| = 13 / 8 by norm_num, |
| show (7 / 4 : ℝ) / (13 / 8) = 14 / 13 by norm_num] at hj |
| exact hj.trans (div_le_div_of_nonneg_right hlogratio hlog.le) |
|
|
| end Ostmann |
|
|
| end OAI |
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|
