§27.7 Lambert Series as Generating Functions

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

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

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 $\sum _{{n=1}}^{\infty}f(n)\frac{x^{n}}{1-x^{n}}=\sum _{{n=1}}^{\infty}\sum _{{d\divides n}}f(d)x^{n}.$

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

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

27.7.3 $\sum _{{n=1}}^{\infty}\mathop{\mu\/}\nolimits\!\left(n\right)\frac{x^{n}}{1-x^{n}}=x,$

Symbols:: $\mathop{\mu\/}\nolimits\!\left(n\right)$ : 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

27.7.4 $\sum _{{n=1}}^{\infty}\mathop{\phi\/}\nolimits\!\left(n\right)\frac{x^{n}}{1-x^{n}}=\frac{x}{(1-x)^{2}},$

Symbols:: $\mathop{\phi _{{k}}\/}\nolimits\!\left(n\right)$ : sum of powers of integers relatively prime to $n$ , $n$ : 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

27.7.5 $\sum _{{n=1}}^{\infty}n^{\alpha}\frac{x^{n}}{1-x^{n}}=\sum _{{n=1}}^{\infty}\mathop{\sigma _{{\alpha}}\/}\nolimits\!\left(n\right)x^{n},$

Symbols:: $\mathop{\sigma _{{\alpha}}\/}\nolimits\!\left(n\right)$ : sum of powers of divisors of $n$ , $n$ : 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

27.7.6 $\sum _{{n=1}}^{\infty}\mathop{\lambda\/}\nolimits\!\left(n\right)\frac{x^{n}}{1-x^{n}}=\sum _{{n=1}}^{\infty}x^{{n^{2}}}.$

Symbols:: $\mathop{\lambda\/}\nolimits\!\left(n\right)$ : Liouville’s function, $n$ : positive integer and $x$ : real number
Permalink:: http://dlmf.nist.gov/27.7.E6
Encodings:: TeX, pMML, png