Higher-order dualities yield ∑ μ(n) ω(n)^k / n = 0 for k ≥ 2 and conditional sums over smallest prime factor p1(n) ≡ j mod ℓ equal to zero or 1/φ(ℓ) for coprime j, ℓ and k ≥ 3.
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Duality Between Prime Factors and The Prime Number Theorem For Arithmetic Progressions -- Higher Order Dualities
Higher-order dualities yield ∑ μ(n) ω(n)^k / n = 0 for k ≥ 2 and conditional sums over smallest prime factor p1(n) ≡ j mod ℓ equal to zero or 1/φ(ℓ) for coprime j, ℓ and k ≥ 3.