OAI.TotientAsymptotic.companion_zero_case
ProvedThe theorem states that, for a natural number k>0 (the hypothesis 0<k is present but unused), if there is no totient value d (a positive integer d equal to φ(n) for some positive n) whose least preimage ell(d), the smallest positive n with φ(n)=d, satisfies k·d < ell(d), then two conclusions hold. First, for every real x>0, the weighted count N(k,x) is zero; N(k,x) counts integers v from 1 to ⌊x⌋ that are totient values with k·x < ell(v) ≤ (k+1)·x. Second, for every s in [0,1), the limiting coefficient A(f_k, s) is zero, where f_k(r)=min(1,(k+1)/r)−min(1,k/r) and A(f,s) is defined as the limit as H tends to infinity of an explicit alternating inclusion-exclusion sum over families of tail witnesses, built from the renewal constants rho, lambda and gamma, which is taken to be a limit value chosen by limUnder.
-- Generated from openai/math @ adc7f1241b42e322a6451854ab7e4b4c146bf78a -- Source: lean/ComparatorChallenges/TotientCompanionZero.lean; bytes 3143..3507 -- Kind: theorem; original declaration names and bodies preserved. -- Source groups are independent. Target: Lean 4.33.1; see compilation.json. import Mathlib import Definitions.Def_TotientCompanionZero namespace OAI noncomputable section open scoped BigOperators Topology open Filter namespace TotientAsymptotic
/-- Without a totient value whose least preimage exceeds its prescribed multiple,
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