The inverse summation of coprime couples
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(107). Give your answer rounded to four decimal places.