PBK's sum of LCM

\[\sum\limits_{i=1}^N\sum\limits_{j=1}^M\frac{a_ia_j}{\gcd(a_i,a_j)} \]

\[\sum\limits_{d=1}^\infty\frac 1 d\sum\limits_{i=1}^N\sum\limits_{j=1}^Ma_ia_j[\gcd(a_i,a_j)=d] \]

\[\sum\limits_{d=1}^\infty\frac 1 d\sum\limits_{i=1}^Na_i\sum\limits_{j=1}^Ma_j[\gcd(a_i,a_j)=d] \]

\[\sum\limits_{d=1}^\infty\frac 1 d\sum\limits_{d|x}\mu(\frac x d)(\sum\limits_{x|a_i}a_i)^2 \]

\[\sum\limits_{d=1}^\infty\frac 1 d\sum\limits_{d|x}\mu(\frac x d)s(x)^2 \]

1. \[\sum\limits_{T=1}^{lim}\mu(T)\sum\limits_{d=1}^{\lfloor\frac{lim}T\rfloor}\frac{s(dT)^2}d \]

没什么用。

2. \[\sum\limits_{x=1}^{lim}s(x)^2\sum\limits_{d|x}\mu(d)\times\frac d x \]

\[\sum\limits_{x=1}^{lim}\frac{s(x)^2}x\sum\limits_{d|x}\mu(d)\times d \]