27 Functions of Number Theory Multiplicative Number Theory 27.6 Divisor Sums 27.8 Dirichlet Characters

§27.7 Lambert Series as Generating Functions

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Keywords:: Lambert series, number theory, number-theoretic functions
Notes:: See Apostol (1990, pp. 24–25).
Permalink:: http://dlmf.nist.gov/27.7
See also:: Annotations for Ch.27

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

27.6 Divisor Sums 27.8 Dirichlet Characters

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§27.7 Lambert Series as Generating Functions

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