## The inverse summation of coprime couples

### Problem 441

Published on Sunday, 20th October 2013, 10:00 am; Solved by 220; Difficulty rating: 65%
For an integer `M`, we define R(`M`) as the sum of 1/(`p`·`q`) for all the integer pairs `p` and `q` which satisfy all of these conditions:

- 1 ≤
`p`<`q`≤`M` -
`p`+`q`≥`M` -
`p`and`q`are coprime.

We also define S(`N`) as the sum of R(`i`) for 2 ≤ `i` ≤ `N`.

We can verify that S(2) = R(2) = 1/2, S(10) ≈ 6.9147 and S(100) ≈ 58.2962.

Find S(10^{7}). Give your answer rounded to four decimal places.