Interesting identity concerning the sum of remainders of n mod k

Doing some analyses on the numbers' plane I realized the existence of some very nice identity concerning the remainders of the division of given n by all the numbers smaller than n.

The sum of these remainders divided by the number of unities within given n tends to 2 - zeta(2)

what's tantamount to

 and going further

And here's how in a very interesting way the value of this first limit tends to the target value on the way to n

It occurs that this identity can be derived from the other, very beautiful one:

And this is how behaves the function showing the sum of remainders and sum of divisors in compare to the function 2n-1.

And the function showing the difference between n^2 and sum of sums of remainders and divisors: