| import OAI.NumberTheory.Ostmann.QuadraticSieve |
| import OAI.NumberTheory.Ostmann.QuadraticSieveSquareLattice |
| import OAI.NumberTheory.Ostmann.QuadraticSieveSquarefreeReindex |
|
|
| namespace OAI |
|
|
| namespace Ostmann.QuadraticSieve |
| open scoped SchwartzMap |
|
|
| theorem tsum_squarefree_coprime_cutoff (K d : ℕ) (f : ℕ → ℂ) : |
| (∑' v : {v : ℕ // Squarefree v}, |
| if v.val ≤ K ∧ Nat.Coprime v.val (2 * d) then f v.val else 0) = |
| ∑ v ∈ oddSquarefreeUpTo K, if Nat.Coprime v d then f v else 0 := by |
| classical |
| change (∑' v : {v : ℕ | Squarefree v}, |
| if v.val ≤ K ∧ Nat.Coprime v.val (2 * d) then f v.val else 0) = _ |
| rw [tsum_subtype {v : ℕ | Squarefree v} (fun v => |
| if v ≤ K ∧ Nat.Coprime v (2 * d) then f v else 0)] |
| rw [tsum_eq_sum (s := oddSquarefreeUpTo K)] |
| · apply Finset.sum_congr rfl |
| intro v hv |
| have h := mem_oddSquarefreeUpTo.mp hv |
| simp [h.2.2.2, h.2.1, Nat.coprime_mul_iff_right, |
| Nat.coprime_two_right, h.2.2.1] |
| · intro v hv |
| by_cases hs : Squarefree v |
| · by_cases h : v ≤ K ∧ Nat.Coprime v (2 * d) |
| · have hc := Nat.coprime_mul_iff_right.mp h.2 |
| have hmem : v ∈ oddSquarefreeUpTo K := mem_oddSquarefreeUpTo.mpr |
| ⟨Nat.pos_of_ne_zero hs.ne_zero, h.1, Nat.coprime_two_right.mp hc.1, hs⟩ |
| exact (hv hmem).elim |
| · simp [hs, h] |
| · simp [hs] |
|
|
| theorem schwartz_summable_positive (W : 𝓢(ℝ, ℂ)) (M : ℝ) (hM : M ≠ 0) : |
| Summable (fun n : {n : ℕ // 0 < n} => W ((n.val : ℝ) / M)) := by |
| have hinj : Function.Injective (fun n : {n : ℕ // 0 < n} => (n.val : ℤ)) := by |
| intro a b h |
| apply Subtype.ext |
| change (a.val : ℤ) = (b.val : ℤ) at h |
| exact Int.ofNat.inj h |
| exact (schwartz_summable_scaled W M hM).comp_injective hinj |
|
|
| theorem schwartz_complement_eq_coprime_lattices (W : 𝓢(ℝ, ℂ)) |
| (M : ℝ) (hM : 0 < M) (K d q : ℕ) [NeZero d] [NeZero q] : |
| (∑' n : {n : ℕ // 0 < n}, if squarefreeKernel n.val ≤ K then |
| (if Odd n.val then (jacobiSym (n.val : ℤ) (d ^ 2 * q) : ℂ) * |
| W ((n.val : ℝ) / M) else 0) else 0) = |
| (1 / 2 : ℂ) * ∑ v ∈ oddSquarefreeUpTo K, if Nat.Coprime v d then |
| (jacobiSym (v : ℤ) q : ℂ) * |
| ∑' m : ℤ, if Nat.Coprime m.natAbs (2 * q * d) then |
| squarePullback W 1 (by norm_num) ((m : ℝ) / Real.sqrt (M / v)) else 0 else 0 := by |
| have hk : 2 ≤ 2 * q * d := by |
| have hq := Nat.pos_of_neZero q |
| have hd := Nat.pos_of_neZero d |
| nlinarith |
| rw [odd_kernel_cutoff_reindex (fun n => W ((n : ℝ) / M)) |
| (schwartz_summable_positive W M hM.ne') K d q] |
| calc |
| _ = ∑' v : {v : ℕ // Squarefree v}, if v.val ≤ K ∧ Nat.Coprime v.val (2 * d) then |
| (1 / 2 : ℂ) * ((jacobiSym (v.val : ℤ) q : ℂ) * |
| ∑' m : ℤ, if Nat.Coprime m.natAbs (2 * q * d) then |
| squarePullback W 1 (by norm_num) ((m : ℝ) / Real.sqrt (M / v.val)) else 0) else 0 := by |
| apply tsum_congr |
| intro v |
| split_ifs with hv |
| · have hlat := positive_coprime_square_lattice W M hM v.val |
| (Nat.pos_of_ne_zero v.property.ne_zero) (2 * q * d) hk |
| push_cast |
| rw [hlat] |
| ring |
| · rfl |
| _ = (1 / 2 : ℂ) * ∑' v : {v : ℕ // Squarefree v}, |
| if v.val ≤ K ∧ Nat.Coprime v.val (2 * d) then |
| (jacobiSym (v.val : ℤ) q : ℂ) * |
| ∑' m : ℤ, if Nat.Coprime m.natAbs (2 * q * d) then |
| squarePullback W 1 (by norm_num) ((m : ℝ) / Real.sqrt (M / v.val)) else 0 else 0 := by |
| rw [← tsum_mul_left] |
| apply tsum_congr |
| intro v |
| split_ifs <;> simp |
| _ = _ := congrArg (fun z : ℂ => (1 / 2 : ℂ) * z) |
| (tsum_squarefree_coprime_cutoff K d (fun v => (jacobiSym (v : ℤ) q : ℂ) * |
| ∑' m : ℤ, if Nat.Coprime m.natAbs (2 * q * d) then |
| squarePullback W 1 (by norm_num) ((m : ℝ) / Real.sqrt (M / v)) else 0)) |
|
|
| end Ostmann.QuadraticSieve |
|
|
| end OAI |
|
|
