Let , for each d|n, define x,n)=d\}" alt="N_d=\{x^2x,n)=d\}" />, then is a partion on N.

And for each , Since , By the definition of S(.), the sum of elements in is ,

Thus,the first equality of

is proved.

When d exhaust all the divisors of n, so do , Thus the second equality obviously hold.