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27 Functions of Number TheoryMultiplicative Number Theory

§27.7 Lambert Series as Generating Functions

Lambert series have the form

27.7.1 n=1f(n)xn1-xn.

If |x|<1, then the quotient xn/(1-xn) is the sum of a geometric series, and when the series (27.7.1) converges absolutely it can be rearranged as a power series:

27.7.2 n=1f(n)xn1-xn=n=1d|nf(d)xn.

Again with |x|<1, special cases of (27.7.2) include:

27.7.3 n=1μ(n)xn1-xn =x,
27.7.4 n=1ϕ(n)xn1-xn =x(1-x)2,
27.7.5 n=1nαxn1-xn =n=1σα(n)xn,
27.7.6 n=1λ(n)xn1-xn =n=1xn2.