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a72760f | 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 | 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
/-- A logarithmic zero count, including analytic multiplicity, on each local disk. -/
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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