e=μ∗Ilarge e=mu*Ie=μ∗I φ∗I=id→φ=μ∗idlarge varphi*I=id\ ovarphi=mu*idφ∗I=id→φ=μ∗id σ0=I∗I→I=μ∗σ0large sigma_0=I*I\ o I=mu*sigma_0σ0=I∗I→I=μ∗σ0
∑d∣nμ(d)=[n=1]large sum_{d|n}mu(d)=[n=1]d∣n∑μ(d)=[n=1] ∑d∣nφ(d)=nlarge sum_{d|n}varphi(d)=nd∣n∑φ(d)=n ∑d∣nμ(nd)⋅d=φ(n)∑d∣nμ(d)⋅nd=φ(n)large sum_{d|n}mu(frac nd)cdot d=varphi(n)\sum_{d|n}mu(d)cdot frac nd=varphi(n)d∣n∑μ(dn)⋅d=φ(n)d∣n∑μ(d)⋅dn=φ(n) σ0(i⋅j)=∑x∣i∑y∣j[(x,y)=1]large sigma_0(icdot j)=sum_{x|i}sum_{y|j}[(x,y)=1]σ0(i⋅j)=x∣i∑y∣j∑[(x,y)=1] σ1(i⋅j)=∑x∣i∑y∣jx⋅j/y[(x,y)=1]large sigma_1(icdot j)=sum_{x|i}sum_{y|j}xcdot j/y[(x,y)=1]σ1(i⋅j)=x∣i∑y∣j∑x⋅j/y[(x,y)=1]