| import OAI.NumberTheory.TwoPointCorrelations.RetainedDivisorMass |
| import OAI.NumberTheory.TwoPointCorrelations.Basic |
|
|
| namespace OAI |
|
|
| /-! Exact expansion of the center-band factors. Each nonempty choice of |
| constant factors leaves an ordinary divisibility condition for the retained |
| prime product, and carries precisely one reciprocal removed-prime product. -/ |
|
|
| namespace TwoPointCorrelations |
|
|
| open Finset |
| open scoped Classical |
|
|
| noncomputable def natDivisibilityIndicator (d n : ℕ) : ℂ := if d ∣ n then 1 else 0 |
|
|
| lemma primeProduct_dvd_iff (S : Finset ℕ) (hS : ∀ p ∈ S, Nat.Prime p) (n : ℕ) : |
| (∏ p ∈ S, p) ∣ n ↔ ∀ p ∈ S, p ∣ n := by |
| constructor |
| · intro h p hp |
| exact (dvd_prod_of_mem (fun p : ℕ => p) hp).trans h |
| · intro h |
| by_cases hn : n = 0 |
| · simp [hn] |
| · have hsub : S ⊆ n.primeFactors := fun p hp => (hS p hp).mem_primeFactors (h p hp) hn |
| exact (prod_dvd_prod_of_subset S n.primeFactors (fun p : ℕ => p) hsub).trans |
| (Nat.prod_primeFactors_dvd n) |
|
|
| lemma natDivisibilityIndicator_product (S : Finset ℕ) |
| (hS : ∀ p ∈ S, Nat.Prime p) (n : ℕ) : |
| (∏ p ∈ S, natDivisibilityIndicator p n) = |
| natDivisibilityIndicator (∏ p ∈ S, p) n := by |
| by_cases hd : (∏ p ∈ S, p) ∣ n |
| · have hp := (primeProduct_dvd_iff S hS n).mp hd |
| unfold natDivisibilityIndicator |
| simp only [hd, ite_true] |
| exact prod_eq_one (fun p hpS => ite_eq_left (hp p hpS)) |
| · have hn : ¬∀ p ∈ S, p ∣ n := fun h => hd ((primeProduct_dvd_iff S hS n).mpr h) |
| push Not at hn |
| obtain ⟨p, hp, hpn⟩ := hn |
| have hz : (∏ p ∈ S, natDivisibilityIndicator p n) = 0 := |
| prod_eq_zero hp (by simp [natDivisibilityIndicator, hpn]) |
| rw [hz] |
| simp only [natDivisibilityIndicator, hd, ite_false] |
|
|
| theorem centerBand_expansion (S : Finset ℕ) (hS : ∀ p ∈ S, Nat.Prime p) |
| (θ : ℂ) (n : ℕ) : |
| (∏ p ∈ S, (natDivisibilityIndicator p n - θ / (p : ℂ))) = |
| ∑ W ∈ S.powerset, ((-θ) ^ W.card / ((∏ p ∈ W, p : ℕ) : ℂ)) * |
| natDivisibilityIndicator (∏ p ∈ S \ W, p) n := by |
| rw [prod_sub] |
| apply sum_congr rfl |
| intro W _ |
| rw [natDivisibilityIndicator_product (S \ W) |
| (fun p hp => hS p (mem_sdiff.mp hp).1) n] |
| rw [prod_div_distrib, prod_const, ← Nat.cast_prod, neg_pow] |
| ring |
|
|
| /-- A nonempty removed-prime set contributes a nontrivial factor `w>1`. -/ |
| lemma primeSubset_product_one_lt (W : Finset ℕ) (hW : ∀ p ∈ W, Nat.Prime p) |
| (hne : W.Nonempty) : 1 < ∏ p ∈ W, p := by |
| obtain ⟨p, hp⟩ := hne |
| have hpos : 0 < ∏ p ∈ W, p := prod_pos (fun q hq => (hW q hq).pos) |
| have hle : p ≤ ∏ q ∈ W, q := Nat.le_of_dvd hpos (dvd_prod_of_mem (fun q : ℕ => q) hp) |
| exact (hW p hp).one_lt.trans_le hle |
|
|
| lemma primeSubset_factorization (S W : Finset ℕ) (hW : W ⊆ S) : |
| (∏ p ∈ S \ W, p) * (∏ p ∈ W, p) = ∏ p ∈ S, p := by |
| rw [← prod_union sdiff_disjoint, sdiff_union_of_subset hW] |
|
|
| end TwoPointCorrelations |
|
|
| end OAI |
|
|
