I am going through the chapter "An exponential sum formed with primes" from the book Multiplicative Number Theory, Davenport. However I am unable to understand the following inequality($\ll$).
$$M + \sum_{1 \leq m \leq N/M} \min\big(\frac{N}{m}, \frac{1}{\|m \alpha\|}\big) \ll\Big( M +\frac{N}M+\frac{N}{q}+q\Big)(\log qN)$$
What I know is $$\sum_{t \leq T} \min\Big(\frac{N}{t}, \frac{1}{\|t \alpha\|}\Big) \ll \Big(\frac Nq+T+q\Big)\log(2qT)$$
I am also uploading the image from the book for clarity