| import OAI.NumberTheory.TwoPoint.Fourier.MajorArcRationalSharp |
| import OAI.NumberTheory.TwoPoint.Fourier.MajorArcReduction |
|
|
| namespace OAI |
|
|
| /-! Preserve the square-root modulus loss through partial summation. |
| The estimates for shorter intervals remain explicit. -/ |
| namespace TwoPointCorrelations |
|
|
| open Finset |
|
|
| theorem major_arc_typical_perturbed_sharp {ι : Type*} (J : Finset ι) |
| (P : ι → Finset ℕ) (hP : ∀ j ∈ J, ∀ p ∈ P j, p.Prime) |
| (F : ℕ → ℂ) (hFm : ∀ a b, 0 < a → 0 < b → F (a*b)=F a*F b) (hF : OneBounded F) |
| (q : ℕ) (hq : 0 < q) (r : ℤ) (X H : ℕ) (hH : 0 < H) |
| (A : ℝ) (hA : 0 ≤ A) (β : ℝ) |
| (havoid : ∀ d ∈ q.divisors, mrtPrimeAvoids (J.biUnion P) d) |
| (hshort : ∀ h : ℕ, 1 ≤ h → h ≤ H → |
| ∀ d : {d : ℕ // d ∈ q.divisors}, ∀ χ : DirichletCharacter ℂ (q/d.val), |
| (d.val:ℝ)*shortExponentialIntegral (mrtTypicalCoefficient J P (twistByCharacter F χ)) |
| (X/d.val+1) (h/d.val+1) 0 ≤ A/(d.val:ℝ)) : |
| shortExponentialIntegral (mrtTypicalCoefficient J P F) X H ((r:ℝ)/q+β) ≤ |
| (1+2*Real.pi*|β| *H)*Real.sqrt (q:ℝ)*((1+Real.log q)*A+(q:ℝ)*X) := by |
| have hq1 : (1:ℝ) ≤ q := by exact_mod_cast hq |
| have hK : 0 ≤ Real.sqrt (q:ℝ)*((1+Real.log q)*A+(q:ℝ)*X) := by |
| have hl := Real.log_nonneg hq1 |
| positivity |
| have hh := major_arc_integral_perturbation_uniform (mrtTypicalCoefficient J P F) |
| X H hH ((r:ℝ)/q) β _ hK (fun h hh hhH => |
| major_arc_typical_rational_sharp J P hP F hFm hF q hq r X h A hA havoid (hshort h hh hhH)) |
| simpa only [mul_assoc] using hh |
|
|
| lemma major_arc_twist_complete {q : ℕ} (F : ℕ → ℂ) |
| (hF : ∀ a b, 0 < a → 0 < b → F (a*b)=F a*F b) |
| (χ : DirichletCharacter ℂ q) : |
| ∀ a b, 0 < a → 0 < b → twistByCharacter F χ (a*b)=twistByCharacter F χ a*twistByCharacter F χ b := by |
| intro a b ha hb |
| simp only [twistByCharacter,hF a b ha hb,Nat.cast_mul,map_mul] |
| ring |
|
|
| lemma major_arc_divisor_distance {F : ℕ → ℂ} {X H : ℕ} {M : ℝ} |
| (hd : MRTDistanceLowerBound F X H M) (q : ℕ) (hq : 0 < q) |
| (hqmax : (q:ℝ) ≤ mrtModulusCutoff X H) |
| (d : {d : ℕ // d ∈ q.divisors}) (χ : DirichletCharacter ℂ (q/d.val)) |
| (t : ℝ) (ht : |t| ≤ X) : |
| M ≤ squaredDistance (twistByCharacter F χ) (mrtArchimedeanTwist t) X := by |
| apply hd.major_arc_character (q/d.val) (major_arc_divisor_quotient_pos hq d) _ χ t ht |
| exact (show ((q/d.val:ℕ):ℝ) ≤ q by exact_mod_cast Nat.div_le_self q d.val).trans hqmax |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
