Proves S_1^φ(x) ≪ x exp{-(1/2-o(1))√(log x log_2 x)} and a decomposition S_h^φ(x) = D_{h,>Y_J}^φ(x) + O(x exp{-√J G + o(V)}) for 1 ≤ h ≤ exp{G/√J} using smooth-totient theorems and supplier systems.
Ford,Solutions of φ(n) = φ(n + k)and σ(n) = σ(n + k), Int
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Rank Amplification for Shifted Equal Values of Euler's Totient Function
Proves S_1^φ(x) ≪ x exp{-(1/2-o(1))√(log x log_2 x)} and a decomposition S_h^φ(x) = D_{h,>Y_J}^φ(x) + O(x exp{-√J G + o(V)}) for 1 ≤ h ≤ exp{G/√J} using smooth-totient theorems and supplier systems.