Let

$ $ T(N) = \begin{cases}1 & \text{if } N = 1\ T(\varphi(N)) + 2T(\sqrt{N}) + \lg(\varphi(N))^3 & \text{otherwise} \end{cases}$ $

where $ \varphi(N)$ is Euler’s totient function. My objective is to find an upper bound so that I can apply the Master Theorem and find a closed-form formula.