# Lambert series

For generalized Lambert series, see Appell–Lerch sum.
Function ${\displaystyle S(q)=\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{n}}}}$, represented as a Matplotlib plot, using a version of the Domain coloring method[1]

In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form

${\displaystyle S(q)=\sum _{n=1}^{\infty }a_{n}{\frac {q^{n}}{1-q^{n}}}.}$

It can be resummed formally by expanding the denominator:

${\displaystyle S(q)=\sum _{n=1}^{\infty }a_{n}\sum _{k=1}^{\infty }q^{nk}=\sum _{m=1}^{\infty }b_{m}q^{m}}$

where the coefficients of the new series are given by the Dirichlet convolution of an with the constant function 1(n) = 1:

${\displaystyle b_{m}=(a*1)(m)=\sum _{n\mid m}a_{n}.\,}$

This series may be inverted by means of the Möbius inversion formula, and is an example of a Möbius transform.

## Examples

Since this last sum is a typical number-theoretic sum, almost any natural multiplicative function will be exactly summable when used in a Lambert series. Thus, for example, one has

${\displaystyle \sum _{n=1}^{\infty }q^{n}\sigma _{0}(n)=\sum _{n=1}^{\infty }{\frac {q^{n}}{1-q^{n}}}}$

where ${\displaystyle \sigma _{0}(n)=d(n)}$ is the number of positive divisors of the number n.

For the higher order sigma functions, one has

${\displaystyle \sum _{n=1}^{\infty }q^{n}\sigma _{\alpha }(n)=\sum _{n=1}^{\infty }{\frac {n^{\alpha }q^{n}}{1-q^{n}}}}$

where ${\displaystyle \alpha }$ is any complex number and

${\displaystyle \sigma _{\alpha }(n)=({\textrm {Id}}_{\alpha }*1)(n)=\sum _{d\mid n}d^{\alpha }\,}$

is the divisor function.

Lambert series in which the an are trigonometric functions, for example, an = sin(2n x), can be evaluated by various combinations of the logarithmic derivatives of Jacobi theta functions.

Other Lambert series include those for the Möbius function ${\displaystyle \mu (n)}$:

${\displaystyle \sum _{n=1}^{\infty }\mu (n)\,{\frac {q^{n}}{1-q^{n}}}=q.}$

For Euler's totient function ${\displaystyle \varphi (n)}$:

${\displaystyle \sum _{n=1}^{\infty }\varphi (n)\,{\frac {q^{n}}{1-q^{n}}}={\frac {q}{(1-q)^{2}}}.}$

For Liouville's function ${\displaystyle \lambda (n)}$:

${\displaystyle \sum _{n=1}^{\infty }\lambda (n)\,{\frac {q^{n}}{1-q^{n}}}=\sum _{n=1}^{\infty }q^{n^{2}}}$

with the sum on the right similar to the Ramanujan theta function.

## Alternate form

Substituting ${\displaystyle q=e^{-z}}$ one obtains another common form for the series, as

${\displaystyle \sum _{n=1}^{\infty }{\frac {a_{n}}{e^{zn}-1}}=\sum _{m=1}^{\infty }b_{m}e^{-mz}}$

where

${\displaystyle b_{m}=(a*1)(m)=\sum _{d\mid m}a_{d}\,}$

as before. Examples of Lambert series in this form, with ${\displaystyle z=2\pi }$, occur in expressions for the Riemann zeta function for odd integer values; see Zeta constants for details.

## Current usage

In the literature we find Lambert series applied to a wide variety of sums. For example, since ${\displaystyle q^{n}/(1-q^{n})=\mathrm {Li} _{0}(q^{n})}$ is a polylogarithm function, we may refer to any sum of the form

${\displaystyle \sum _{n=1}^{\infty }{\frac {\xi ^{n}\,\mathrm {Li} _{u}(\alpha q^{n})}{n^{s}}}=\sum _{n=1}^{\infty }{\frac {\alpha ^{n}\,\mathrm {Li} _{s}(\xi q^{n})}{n^{u}}}}$

as a Lambert series, assuming that the parameters are suitably restricted. Thus

${\displaystyle 12\left(\sum _{n=1}^{\infty }n^{2}\,\mathrm {Li} _{-1}(q^{n})\right)^{\!2}=\sum _{n=1}^{\infty }n^{2}\,\mathrm {Li} _{-5}(q^{n})-\sum _{n=1}^{\infty }n^{4}\,\mathrm {Li} _{-3}(q^{n}),}$

which holds for all complex q not on the unit circle, would be considered a Lambert series identity. This identity follows in a straightforward fashion from some identities published by the Indian mathematician S. Ramanujan. A very thorough exploration of Ramanujan's works can be found in the works by Bruce Berndt.