μ ∗ 1 = e μ * 1 = e μ∗1=e φ ∗ 1 = i d ⇒ φ = μ ∗ i d φ * 1 = id \Rightarrow φ = μ * id φ∗1=id⇒φ=μ∗id d = 1 ∗ 1 ⇒ 1 = μ ∗ d d = 1* 1 \Rightarrow 1 = μ * d d=1∗1⇒1=μ∗d 其中: ∑ d ∣ n μ ( d ) = [ n = = 1 ] \sum_{d|n} μ(d) = {[n==1]} d∣n∑μ(d)=[n==1] ∑ d ∣ n φ ( d ) = n \sum_{d|n} φ(d) = {n} d∣n∑φ(d)=n ∑ d ∣ n μ ( n / d ) ∗ d = φ ( n ) \sum_{d|n} μ(n/d)*d = {φ(n)} d∣n∑μ(n/d)∗d=φ(n) ∑ d ∣ n μ ( n / d ) ∗ d = φ ( n ) \sum_{d|n} μ(n/d)*d = {φ(n)} d∣n∑μ(n/d)∗d=φ(n) d ( i , j ) = ∑ p ∣ i ∑ q ∣ j [ g c d ( p , q ) = 1 ] d(i,j) = \sum_{p|i} \sum_{q|j} [gcd(p,q)=1] d(i,j)=p∣i∑q∣j∑[gcd(p,q)=1]
e [ n ] = [ n = = 1 ] e[n]=[n==1] e[n]=[n==1] d ( n ) d(n) d(n)为n的因数个数 i d ( n ) = n id(n)=n id(n)=n
