| import Mathlib |
|
|
| namespace OAI |
|
|
| /-! The no-seed alternative of the weighted totient companion theorem. -/ |
|
|
| noncomputable section |
| open scoped BigOperators Topology |
| open Filter |
|
|
| namespace TotientAsymptotic |
|
|
| def IsTotient (v : ℕ) : Prop := ∃ n : ℕ, 0 < n ∧ n.totient = v |
|
|
| def ell (v : ℕ) : ℕ := by |
| classical |
| exact if h : IsTotient v then Nat.find h else 0 |
|
|
| def a (j : ℕ) : ℝ := (j + 1 : ℝ) * Real.log (j + 1) - j * Real.log j - 1 |
|
|
| def RenewalRootSet : Set ℝ := |
| {z | 0 < z ∧ z < 1 ∧ (∑' j : ℕ, a (j+1) * z^(j+1)) = 1} |
|
|
| def rho : ℝ := sInf RenewalRootSet |
|
|
| def lam : ℝ := Real.log (1 / rho) |
|
|
| def gamma : ℝ := (∑' j : ℕ, (j+1 : ℝ) * a (j+1) * rho^(j+1))⁻¹ |
|
|
| def P (H : ℕ) : ℕ := ⌊Real.log (Real.log (H : ℝ))⌋₊ |
| def alpha (s : ℝ) : ℝ := lam * Real.exp (lam * s) |
|
|
| def largestPrimeFactor (n : ℕ) : ℕ := max 1 (n.primeFactors.sup id) |
|
|
| structure TailDatum (H : ℕ) where |
| Q : Fin H → ℕ |
| cofactor : ℕ |
|
|
| def tailPrime {H : ℕ} (η : TailDatum H) (h : ℕ) : ℕ := |
| if hH : h < H then η.Q ⟨h, hH⟩ else 1 |
|
|
| def tailLog {H : ℕ} (η : TailDatum H) (h : ℕ) : ℝ := |
| Real.log (Real.log (tailPrime η h : ℝ)) |
|
|
| def IsWitness (H : ℕ) (s : ℝ) (η : TailDatum H) : Prop := |
| (∀ h : Fin H, h.val < P H → η.Q h = 1) ∧ |
| 0 < η.cofactor ∧ |
| (∀ h ∈ Finset.Ico (P H) H, (tailPrime η h).Prime ∧ |
| (9 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ≤ tailLog η h ∧ |
| tailLog η h ≤ (11 / 10 : ℝ) * alpha s * h * (rho^h)⁻¹ ∧ |
| (∑ l ∈ Finset.Ico (P H) h, a (h-l) * tailLog η l) ≤ |
| (1 + (1 / 10000 : ℝ) * Real.exp (-(h : ℝ) / 40)) * tailLog η h) ∧ |
| largestPrimeFactor η.cofactor ≤ tailPrime η (P H) ∧ |
| Real.log (η.cofactor : ℝ) ≤ |
| Real.exp (2 * alpha s * (P H : ℝ) * (rho^(P H))⁻¹) |
|
|
| def w {H : ℕ} (η : TailDatum H) : ℕ := |
| η.cofactor * ∏ h ∈ Finset.Ico (P H) H, tailPrime η h |
|
|
| def witnesses (H : ℕ) (s : ℝ) (d : ℕ) : Set (TailDatum H) := |
| {η | IsWitness H s η ∧ (w η).totient = d} |
|
|
| def D {H : ℕ} (h : ℕ) (η : TailDatum H) : ℝ := |
| ∑ l ∈ Finset.Ico (P H) H, a (h-l) * tailLog η l |
|
|
| def maxD {H : ℕ} (h : ℕ) (T : Finset (TailDatum H)) : ℝ := by |
| classical |
| exact if ht : T.Nonempty then T.sup' ht (D h) else 0 |
|
|
| def AH (H : ℕ) (f : ℝ → ℝ) (s : ℝ) : ℝ := by |
| classical |
| exact rho^(H*(H-1)/2) * (gamma / alpha s)^H * |
| ∑ᶠ d : ℕ, |
| if IsTotient d then |
| f ((ell d : ℝ) / d) / d * |
| ∑ᶠ T : Finset (TailDatum H), |
| if T.Nonempty ∧ (↑T : Set (TailDatum H)) ⊆ witnesses H s d then |
| (-1 : ℝ)^(T.card-1) * |
| Real.exp (-(gamma / alpha s) * |
| ∑' n : ℕ, rho^(H+n) * maxD (H+n) T) |
| else 0 |
| else 0 |
|
|
| def A (f : ℝ → ℝ) (s : ℝ) : ℝ := limUnder atTop (fun H : ℕ => AH H f s) |
|
|
| def N (k : ℕ) (x : ℝ) : ℝ := by |
| classical |
| exact (((Finset.Icc 1 ⌊x⌋₊).filter fun v => |
| IsTotient v ∧ (k : ℝ) * x < ell v ∧ (ell v : ℝ) ≤ (k+1 : ℝ) * x).card : ℝ) |
|
|
| def fk (k : ℕ) (r : ℝ) : ℝ := min 1 ((k+1 : ℝ) / r) - min 1 ((k : ℝ) / r) |
|
|
| / |
| both the weighted count and its limiting coefficient vanish. -/ |
| theorem companion_zero_case (k : ℕ) (_hk : 0 < k) |
| (hno : ¬ ∃ d : ℕ, IsTotient d ∧ k*d < ell d) : |
| (∀ x : ℝ, 0 < x → N k x = 0) ∧ |
| (∀ s ∈ Set.Ico (0 : ℝ) 1, A (fk k) s = 0) := by |
| sorry |
|
|
| end TotientAsymptotic |
| end |
|
|
| end OAI |
|
|
