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

§27.7 Lambert Series as Generating Functions

ⓘ

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 $\left|x\right|<1$ , 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 $\left|x\right|<1$ , 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

NIST

Site Privacy Accessibility Privacy Program Copyrights Vulnerability Disclosure No Fear Act Policy FOIA Environmental Policy Scientific Integrity Information Quality Standards Commerce.gov Science.gov USA.gov