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# Lambert series

In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form
$S\left(q\right)=sum_\left\{n=1\right\}^infty a_n frac \left\{q^n\right\}\left\{1-q^n\right\}$

It can be resummed formally by expanding the denominator:

$S\left(q\right)=sum_\left\{n=1\right\}^infty a_n sum_\left\{k=1\right\}^infty q^\left\{nk\right\} = sum_\left\{m=1\right\}^infty b_m q^m$

where the coefficients of the new series are given by the Dirichlet convolution of $\left\{a_n\right\}$ with the constant function $1\left(n\right)=1$:

$b_m = \left(a*1\right)\left(m\right) = sum_\left\{nmid m\right\} a_n ,$

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 multiplicative function will be exactly summable when used in a Lambert series. Thus, for example, one has

$sum_\left\{n=1\right\}^\left\{infty\right\} q^n sigma_0\left(n\right) = sum_\left\{n=1\right\}^\left\{infty\right\} frac\left\{q^n\right\}\left\{1-q^n\right\}$

where $sigma_0\left(n\right)=d\left(n\right)$ is the number of positive divisors of the number $n$.

For the higher order sigma functions, one has

$sum_\left\{n=1\right\}^\left\{infty\right\} q^n sigma_alpha\left(n\right) = sum_\left\{n=1\right\}^\left\{infty\right\} frac\left\{n^alpha q^n\right\}\left\{1-q^n\right\}$
where $alpha$ is any complex number and
$sigma_alpha\left(n\right) = \left(textrm\left\{Id\right\}_alpha*1\right)\left(n\right) = sum_\left\{dmid n\right\} d^alpha ,$
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 $mu\left(n\right)$:

$sum_\left\{n=1\right\}^infty mu\left(n\right),frac\left\{q^n\right\}\left\{1-q^n\right\} = q.$

For Euler's totient function $phi\left(n\right)$:

$sum_\left\{n=1\right\}^infty varphi\left(n\right),frac\left\{q^n\right\}\left\{1-q^n\right\} = frac\left\{q\right\}\left\{\left(1-q\right)^2\right\}.$

For Liouville's function $lambda\left(n\right)$:

$sum_\left\{n=1\right\}^infty lambda\left(n\right),frac\left\{q^n\right\}\left\{1-q^n\right\} =$
sum_{n=1}^infty q^{n^2}

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

## Alternate form

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

$sum_\left\{n=1\right\}^infty frac \left\{a_n\right\}\left\{e^\left\{zn\right\}-1\right\}= - sum_\left\{m=1\right\}^infty b_m e^\left\{-mz\right\}$

where

$b_m = \left(a*1\right)\left(m\right) = sum_\left\{nmid m\right\} a_n,$

as before. Examples of Lambert series in this form, with $z=2pi$, 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 $q^n/\left(1 - q^n \right) = mathrm\left\{Li\right\}_0\left(q^\left\{n\right\}\right)$ is a polylogarithm function, we may refer to any sum of the form

$sum_\left\{n=1\right\}^\left\{infty\right\} frac\left\{xi^n ,mathrm\left\{Li\right\}_u \left(alpha q^n\right)\right\}\left\{n^s\right\} = sum_\left\{n=1\right\}^\left\{infty\right\} frac\left\{alpha^n ,mathrm\left\{Li\right\}_s\left(xi q^n\right)\right\}\left\{n^u\right\}$

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

$12left\left(sum_\left\{n=1\right\}^\left\{infty\right\} n^2 , mathrm\left\{Li\right\}_\left\{-1\right\}\left(q^n\right)right\right)^\left\{!2\right\} = sum_\left\{n=1\right\}^\left\{infty\right\}$
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.