# What’s an upper bound for this recurrence so I can take advantage of the Master Theorem?

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.