Simply note that in the sum each is counted as many times as multiples it has in , which is exactly: .
Thus:
Let then of course: . By the definition of the floor function:
Hence: and assertion is proven.
- If we also had then (little O) -
This problem is my own invention. I hope you enjoy it.
Suppose the infinite set has zero density in (so that the proportion of elements of less than goes to 0 as ).
Let be the number of elements of which divide .
Show that .
For instance the average number of prime divisors of is asymptotically equal to . The average number of square divisors is asymptotically equal to .