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27 Functions of Number TheoryMultiplicative Number Theory

§27.3 Multiplicative Properties

Except for ν(n), Λ(n), pn, and π(x), the functions in §27.2 are multiplicative, which means f(1)=1 and

27.3.1 f(mn)=f(m)f(n),
(m,n)=1.

If f is multiplicative, then the values f(n) for n>1 are determined by the values at the prime powers. Specifically, if n is factored as in (27.2.1), then

27.3.2 f(n)=r=1ν(n)f(prar).

In particular,

27.3.3 ϕ(n) =np|n(1-p-1),
27.3.4 Jk(n) =nkp|n(1-p-k),
27.3.5 d(n) =r=1ν(n)(1+ar),
27.3.6 σα(n) =r=1ν(n)prα(1+ar)-1prα-1,
α0.

Related multiplicative properties are

27.3.7 σα(m)σα(n)=d|(m,n)dασα(mnd2),
27.3.8 ϕ(m)ϕ(n)=ϕ(mn)ϕ((m,n))/(m,n).

A function f is completely multiplicative if f(1)=1 and

27.3.9 f(mn)=f(m)f(n),
m,n=1,2,.

Examples are 1/n and λ(n), and the Dirichlet characters, defined in §27.8.

If f is completely multiplicative, then (27.3.2) becomes

27.3.10 f(n)=r=1ν(n)(f(pr))ar.