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 $\sum_{n=1}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}.$

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Symbols:: : positive integer and : 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 $\sum_{n=1}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\sum_{d% \mathbin{|}n}f(d)x^{n}.$

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Symbols:: : positive integer, : positive integer and : 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 $\sum_{n=1}^{\infty}\mu\left(n\right)\frac{x^{n}}{1-x^{n}}=x,$

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Symbols:: $\mu\left(\NVar{n}\right)$ : Möbius function, : positive integer and : 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 $\sum_{n=1}^{\infty}\phi\left(n\right)\frac{x^{n}}{1-x^{n}}=\frac{x}{(1-x)^{2}},$

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Symbols:: $\phi\left(\NVar{n}\right)$ : Euler’s totient, : positive integer and : 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 $\sum_{n=1}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{\infty}\sigma_{% \alpha}\left(n\right)x^{n},$

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Symbols:: $\sigma_{\NVar{\alpha}}\left(\NVar{n}\right)$ : sum of powers of divisors of a number, : positive integer and : 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 $\sum_{n=1}^{\infty}\lambda\left(n\right)\frac{x^{n}}{1-x^{n}}=\sum_{n=1}^{% \infty}x^{n^{2}}.$

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Symbols:: $\lambda\left(\NVar{n}\right)$ : Liouville’s function, : positive integer and : 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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