| import OAI.NumberTheory.PrimeGaps.RieszDecay | |
| namespace OAI | |
| namespace LargePrimeGaps | |
| open Complex Real Set MeasureTheory Filter Topology | |
| open Complex Filter Topology | |
| open Complex Filter Topology MeasureTheory | |
| open scoped FourierTransform | |
| open Complex Real Set MeasureTheory Filter Topology | |
| open Complex Real Set MeasureTheory Filter Topology | |
| open Complex Real Set MeasureTheory Filter Topology | |
| open scoped FourierTransform | |
| open Complex Real Set MeasureTheory Filter Topology | |
| open Complex Real Set MeasureTheory Filter Topology | |
| open Complex Real Set MeasureTheory Filter Topology | |
| lemma primeCharacterError_log_bound (A B : ℝ) (hA : 0<A) (hB : 0<B) : | |
| ∃ M : ℝ, 0<M ∧ ∀ (q : ℕ) [NeZero q] (χ : DirichletCharacter ℂ q), χ.IsPrimitive → | |
| ∀ x : ℝ, 2≤x → 1≤Real.log x → (Real.log x)^(2*A+2)≤x → | |
| (q:ℝ)≤(Real.log x)^B → | |
| ‖primeCharacterError x χ‖≤M*x*(Real.log x)^(-A) := by | |
| obtain ⟨D,hD,hb⟩ := primeWeighted_error_bound (B+2*A+2) | |
| refine ⟨15*D+8,by positivity,?_⟩ | |
| intro q _ χ hp x hx hL hpower hq | |
| let L := Real.log x | |
| let K := A+2 | |
| let a := B+2*A+2 | |
| let Q := ((q:ℝ)+2)^(1/2:ℝ) | |
| let h := x/L^K | |
| have hx0 : 0<x := by linarith | |
| have hx1 : 1<x := by linarith | |
| have hL0 : 0<L := lt_of_lt_of_le zero_lt_one hL | |
| have hK : 0<K := by dsimp [K]; linarith | |
| have ha : 0<a := by dsimp [a]; linarith | |
| have hpK : 0<L^K := Real.rpow_pos_of_pos hL0 _ | |
| have hh0 : 0<h := div_pos hx0 hpK | |
| have hK1 : 1≤L^K := Real.one_le_rpow hL hK.le | |
| have hhx : h≤x := (div_le_self hx0.le hK1) | |
| have hKh : L^K≤x := by | |
| apply (Real.rpow_le_rpow_of_exponent_le hL (show K≤2*A+2 by dsimp [K]; linarith)).trans | |
| exact hpower | |
| have hh1 : 1≤h := (le_div_iff₀ hpK).mpr (by simpa using hKh) | |
| have hyx : x≤x+h := by linarith | |
| have hy : 1<x+h := by linarith | |
| have hy2 : x+h≤2*x := by linarith | |
| have hLy : L≤Real.log (x+h) := Real.log_le_log hx0 hyx | |
| have hLy1 : 1≤Real.log (x+h) := hL.trans hLy | |
| have hLy2 : Real.log (x+h)≤2*L := by | |
| calc | |
| _ ≤ Real.log (x^2) := Real.log_le_log (by linarith) (by nlinarith) | |
| _ = _ := by rw [Real.log_pow]; rfl | |
| have hQ0 : 0≤Q := by dsimp [Q]; positivity | |
| have hQ : Q≤3*L^B := by | |
| have hq2 : 1≤(q:ℝ)+2 := by have := Nat.cast_nonneg (α:=ℝ) q; linarith | |
| have hqpow : Q≤(q:ℝ)+2 := by | |
| simpa only [Real.rpow_one] using | |
| Real.rpow_le_rpow_of_exponent_le hq2 (show (1/2:ℝ)≤1 by norm_num) | |
| have hLB : 1≤L^B := Real.one_le_rpow hL hB.le | |
| linarith | |
| let E (y : ℝ) := ‖weightedPrefix (primeCoefficients χ) y-characterMain χ*(y:ℂ)^2/2‖ | |
| have hEx : E x≤D*Q*x^2*L^(-a)+x := hb q χ hp x hx1 hL | |
| have hEy : E (x+h)≤4*D*Q*x^2*L^(-a)+2*x := by | |
| calc | |
| _ ≤ D*Q*(x+h)^2*(Real.log (x+h))^(-a)+(x+h) := hb q χ hp (x+h) hy hLy1 | |
| _ ≤ D*Q*(2*x)^2*L^(-a)+2*x := by | |
| apply add_le_add _ hy2 | |
| apply mul_le_mul _ (Real.rpow_le_rpow_of_nonpos hL0 hLy (neg_nonpos.mpr ha.le)) (by positivity) (by positivity) | |
| gcongr | |
| _ = _ := by ring | |
| have hcoeff (n : ℕ) (hn : n∈Finset.Ioc ⌊x⌋₊ ⌊x+h⌋₊) : | |
| ‖primeCoefficients χ n‖≤2*L := by | |
| have hn0 : 0<(n:ℝ) := by exact_mod_cast (show 0<n by have := (Finset.mem_Ioc.mp hn).1; omega) | |
| have hnx : (n:ℝ)≤x+h := (Nat.le_floor_iff (by linarith : 0≤x+h)).mp (Finset.mem_Ioc.mp hn).2 | |
| exact (primeCoefficients_norm_le χ n).trans ((Real.log_le_log hn0 hnx).trans hLy2) | |
| have heq : primeCharacterError x χ= | |
| (∑ n∈Finset.Ioc 0 ⌊x⌋₊, primeCoefficients χ n)-characterMain χ*(x:ℂ) := by | |
| classical | |
| simp only [primeCharacterError,primeCharacterPrefix,primeCoefficients,characterMain] | |
| split_ifs <;> simp | |
| rw [heq] | |
| have hu := prefix_unsmoothing hx0.le hh0 (show 0≤2*L by positivity) | |
| (primeCoefficients χ) (characterMain χ) hcoeff | |
| change _≤(E (x+h)+E x)/h+(h+1)*(2*L)+‖characterMain χ‖*h/2 at hu | |
| have hδ : ‖characterMain χ‖*h/2≤h := by | |
| have := mul_le_mul_of_nonneg_right (characterMain_norm_le χ) hh0.le | |
| linarith | |
| have htail : (h+1)*(2*L)≤4*h*L := by nlinarith | |
| have herr : (E (x+h)+E x)/h ≤ 15*D*x*L^(-A)+3*x*L^(-A) := by | |
| calc | |
| _ ≤ (5*D*Q*x^2*L^(-a)+3*x)/h := by | |
| apply div_le_div_of_nonneg_right _ hh0.le | |
| linarith | |
| _ ≤ (5*D*(3*L^B)*x^2*L^(-a)+3*x)/h := by gcongr | |
| _ = (15*D*L^B*x^2*L^(-a)+3*x)/h := by ring | |
| _ = 15*D*x*L^(-A)+3*L^K := by | |
| dsimp [h] | |
| rw [div_eq_mul_inv,inv_div] | |
| have hr : L^B*L^(-a)*L^K=L^(-A) := by | |
| rw [←Real.rpow_add hL0,←Real.rpow_add hL0] | |
| congr 1 | |
| dsimp [a,K] | |
| ring | |
| field_simp | |
| linear_combination 15*D*x*hr | |
| _ ≤ _ := by | |
| have ht : L^K≤x*L^(-A) := by | |
| calc | |
| _ = L^(2*A+2)*L^(-A) := by rw [←Real.rpow_add hL0]; congr 1; dsimp [K]; ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_right hpower (by positivity) | |
| nlinarith [ht] | |
| have hsmall : h≤x*L^(-A) := by | |
| dsimp [h] | |
| rw [div_eq_mul_inv,←Real.rpow_neg hL0.le] | |
| exact mul_le_mul_of_nonneg_left | |
| (Real.rpow_le_rpow_of_exponent_le hL (show -K≤-A by dsimp [K]; linarith)) hx0.le | |
| have htail' : 4*h*L≤4*x*L^(-A) := by | |
| calc | |
| _ = 4*x*L^(1-K) := by | |
| dsimp [h] | |
| rw [div_eq_mul_inv,Real.rpow_sub hL0,Real.rpow_one] | |
| ring | |
| _ ≤ _ := mul_le_mul_of_nonneg_left | |
| (Real.rpow_le_rpow_of_exponent_le hL (show 1-K≤-A by dsimp [K]; linarith)) (by positivity) | |
| calc | |
| _ ≤ (E (x+h)+E x)/h+(h+1)*(2*L)+‖characterMain χ‖*h/2 := hu | |
| _ ≤ (15*D*x*L^(-A)+3*x*L^(-A))+4*x*L^(-A)+x*L^(-A) := | |
| add_le_add (add_le_add herr (htail.trans htail')) (hδ.trans hsmall) | |
| _ = _ := by ring | |
| theorem siegelWalfisz : SiegelWalfisz := by | |
| intro A B hA hB | |
| obtain ⟨M,hM,hb⟩ := primeCharacterError_log_bound A B hA hB | |
| have hevent : ∀ᶠ x : ℝ in atTop, 2≤x ∧ 1≤Real.log x ∧ (Real.log x)^(2*A+2)≤x := by | |
| filter_upwards [eventually_ge_atTop (2:ℝ),eventually_ge_atTop (Real.exp 1), | |
| eventually_log_rpow_le_rpow (2*A+2) (by norm_num : (0:ℝ)<1)] with x hx hex hp | |
| refine ⟨hx,?_,by simpa only [Real.rpow_one] using hp⟩ | |
| rw [←Real.log_exp (1:ℝ)] | |
| exact Real.log_le_log (Real.exp_pos _) hex | |
| obtain ⟨x₀,hx₀⟩ := eventually_atTop.mp hevent | |
| refine ⟨M,hM,max x₀ 2,by have := le_max_right x₀ (2:ℝ); linarith,?_⟩ | |
| intro x hx d hd hdlog χ hχ | |
| let : NeZero d := ⟨ne_of_gt hd⟩ | |
| obtain ⟨hx2,hlog,hpower⟩ := hx₀ x ((le_max_left _ _).trans hx) | |
| exact hb d χ hχ x hx2 hlog hpower hdlog | |
| theorem bombieriVinogradov : BombieriVinogradov := | |
| bombieriVinogradov_of_siegelWalfisz siegelWalfisz | |
| theorem main : ∀ C : ℝ, 0 < C → | |
| ∃ c : ℝ, 0 < c ∧ ∃ N₀ : ℕ, ∀ N : ℕ, N₀ ≤ N → | |
| c * (N : ℝ) ≤ ((largeGapIndices C N).card : ℝ) := | |
| index_density_of_bombieriVinogradov bombieriVinogradov | |
| end LargePrimeGaps | |
| end OAI | |