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27 Functions of Number TheoryMultiplicative Number Theory

§27.6 Divisor Sums

Sums of number-theoretic functions extended over divisors are of special interest. For example,

27.6.1 d|nλ(d)={1,n is a square,0,otherwise.

If f is multiplicative, then

27.6.2 d|nμ(d)f(d)=p|n(1-f(p)),

Generating functions, Euler products, and Möbius inversion are used to evaluate many sums extended over divisors. Examples include:

27.6.3 d|n|μ(d)| =2ν(n),
27.6.4 d2|nμ(d) =|μ(n)|,
27.6.5 d|n|μ(d)|ϕ(d) =nϕ(n),
27.6.6 d|nϕk(d)(nd)k=1k+2k++nk,
27.6.7 d|nμ(d)(nd)k=Jk(n),
27.6.8 d|nJk(d)=nk.