tesuto-Mobius

求 egin{equation*}sum_{i=1}^nsum_{j=1}^m[gcd(i,j)=k]end{equation*} 的值.

莫比乌斯反演吧.

egin{align*}
&=sum_{i=1}^{leftlfloorfrac n k ight floor}sum_{j=1}^{leftlfloorfrac m k ight floor}sum_{d|gcd(i,j)=1}mu(d)\
&=sum_{i=1}^{leftlfloorfrac n k ight floor}sum_{j=1}^{leftlfloorfrac m k ight floor}sum_{d|gcd(i,j)=1}mu(d)\
&=sum_{i=1}^{leftlfloorfrac n k ight floor}sum_{j=1}^{leftlfloorfrac m k ight floor}sum_{d|i}sum_{d|j}mu(d)\
&=sum_{i=1}^{leftlfloorfrac n k ight floor}sum_{d|i}sum_{j=1}^{leftlfloorfrac m k ight floor}sum_{d|j}mu(d)\
&=sum_{d=1}^{minleft(leftlfloorfrac n k ight floor,leftlfloorfrac m k ight floor ight)}mu(d)sum_{i=1}^{leftlfloorfrac n k ight floor}sum_{d|i}sum_{j=1}^{leftlfloorfrac m k ight floor}sum_{d|j}1\
&=sum_{d=1}^{minleft(leftlfloorfrac n k ight floor,leftlfloorfrac m k ight floor ight)}mu(d)sum_{i=1}^{leftlfloorfrac n k ight floor}sum_{d|i}1sum_{j=1}^{leftlfloorfrac m k ight floor}sum_{d|j}1\
&=sum_{d=1}^{minleft(leftlfloorfrac n k ight floor,leftlfloorfrac m k ight floor ight)}mu(d)leftlfloorfrac{leftlfloorfrac n k ight floor}d ight floorleftlfloorfrac{leftlfloorfrac m k ight floor}d ight floor\
&=sum_{d=1}^{minleft(leftlfloorfrac n k ight floor,leftlfloorfrac m k ight floor ight)}mu(d)leftlfloorfrac n{kd} ight floorleftlfloorfrac m{kd} ight floor\
end{align*}