[d(ij)=sum_{x|i}sum_{y|j}[gcd(x,y)=1]\
]
[sum_{i=1}^{n}sum_{j=1}^{m}sum_{x|i}sum_{y|j}[gcd(x,y)=1]\
sum_{x=1}^{n}sum_{y=1}^{m}[gcd(x,y)=1]sum_{i=1}^{frac nx}sum_{j=1}^{frac my}1\
sum_{x=1}^{n}sum_{y=1}^{m}(frac nx)(frac my)sum_{p|x,p|y}mu(p)\
sum_{p=1}^{n} mu(p) sum_{x=1}^{frac np}sum_{y=1}^{frac mp}(frac n{xp})(frac m{yp})\
]