For an integer $M$, we define $R(M)$ as the sum of $1/(p\cdot q)$ for all the integer pairs $p$ and $q$ which satisfy all of these conditions:

• $1\le p<q\le M$
• $p+q\ge M$
• $p$ and $q$ are coprime.

We also define $S(N)$ as the sum of $R(i)$ for $2\le i\le N$.
We can verify that $S(2)=R(2)=1/2$, $S(10)\approx 6.9147$ and $S(100)\approx 58.2962$.

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