有大神可以解释一下下面式子的原理吗？
∑i=1A∑j=1B∑k=1Cgcd(i,j,k)×ln(lcm(i,j)gcd(i,j))=∑d=1min(A,B,C)φ(d)∑i=1Ad∑j=1Bd∑k=1Cdln(lcm(id,jd)gcd(id,jd))\sum\limits_{i=1}^A\sum\limits_{j=1}^B\sum\limits_{k=1}^C\gcd(i,j,k)\times \ln(\dfrac{\text{lcm}(i,j)}{\gcd(i,j)})=\sum\limits_{d=1}^{\min(A,B,C)}\varphi(d)\sum\limits_{i=1}^\frac{A}{d}\sum\limits_{j=1}^{\frac{B}{d}}\sum\limits_{k=1}^{\frac{C}{d}} \ln(\dfrac{\text{lcm}(id,jd)}{\gcd(id,jd)})i=1∑Aj=1∑Bk=1∑Cgcd(i,j,k)×ln(gcd(i,j)lcm(i,j))=d=1∑min(A,B,C)φ(d)i=1∑dAj=1∑dBk=1∑dCln(gcd(id,jd)lcm(id,jd))
