See this image (https://en.m.wikipedia.org/wiki/Euler%27s_totient_function#/media/File%3AEulerPhi.svg) from wikipedia. I can make out two lines that have a high density of values along them. The top line consists of the values of the function at the prime numbers, and then the line below it which has about half its slope I'm not sure about. There's also a third less dense line that is about halfway between them. Can anyone tell me why these other two lines have a comparatively high density of values along them? Also, is the maximum of the totient function for values in [1, n] always achieved at a prime? What is a good asymptotic lower bound for the totient function?
Edit: the answers to my second and third question are in the article containing the image. Can anyone explain the lower bound n/loglogn? I know loglogn is the expected number of prime divisors of a randomly chosen element in [1,n].