# §27.7 Lambert Series as Generating Functions

Lambert series have the form

 27.7.1 $\sum_{n=1}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}.$ ⓘ Symbols: $n$: positive integer and $x$: real number Referenced by: §27.7 Permalink: http://dlmf.nist.gov/27.7.E1 Encodings: TeX, pMML, png See also: Annotations for §27.7 and Ch.27

If $|x|<1$, then the quotient $x^{n}/(1-x^{n})$ 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 $\sum_{n=1}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\sum_{d% \mathbin{|}n}f(d)x^{n}.$ ⓘ Symbols: $d$: positive integer, $n$: positive integer and $x$: real number Referenced by: §27.7 Permalink: http://dlmf.nist.gov/27.7.E2 Encodings: TeX, pMML, png See also: Annotations for §27.7 and Ch.27

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

 27.7.3 $\displaystyle\sum_{n=1}^{\infty}\mu\left(n\right)\frac{x^{n}}{1-x^{n}}$ $\displaystyle=x,$ ⓘ Symbols: $\mu\left(\NVar{n}\right)$: Möbius function, $n$: positive integer and $x$: real number A&S Ref: 24.3.1 I.B Permalink: http://dlmf.nist.gov/27.7.E3 Encodings: TeX, pMML, png See also: Annotations for §27.7 and Ch.27 27.7.4 $\displaystyle\sum_{n=1}^{\infty}\phi\left(n\right)\frac{x^{n}}{1-x^{n}}$ $\displaystyle=\frac{x}{(1-x)^{2}},$ ⓘ Symbols: $\phi\left(\NVar{n}\right)$: Euler’s totient, $n$: positive integer and $x$: real number A&S Ref: 24.3.2 I.B Permalink: http://dlmf.nist.gov/27.7.E4 Encodings: TeX, pMML, png See also: Annotations for §27.7 and Ch.27 27.7.5 $\displaystyle\sum_{n=1}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}}$ $\displaystyle=\sum_{n=1}^{\infty}\sigma_{\alpha}\left(n\right)x^{n},$ ⓘ Symbols: $\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)$: sum of powers of divisors of a number, $n$: positive integer and $x$: real number A&S Ref: 24.3.3 I.B Permalink: http://dlmf.nist.gov/27.7.E5 Encodings: TeX, pMML, png See also: Annotations for §27.7 and Ch.27 27.7.6 $\displaystyle\sum_{n=1}^{\infty}\lambda\left(n\right)\frac{x^{n}}{1-x^{n}}$ $\displaystyle=\sum_{n=1}^{\infty}x^{n^{2}}.$ ⓘ Symbols: $\lambda\left(\NVar{n}\right)$: Liouville’s function, $n$: positive integer and $x$: real number Permalink: http://dlmf.nist.gov/27.7.E6 Encodings: TeX, pMML, png See also: Annotations for §27.7 and Ch.27