Lambert series

For generalized Lambert series, see Appell–Lerch sum.

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

$S(q)=\sum_{n=1}^\infty a_n \frac {q^n}{1-q^n}.$

It can be resummed formally by expanding the denominator:

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

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

$b_m = (a*1)(m) = \sum_{n\mid m} a_n. \,$

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

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

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

For the higher order sigma functions, one has

$\sum_{n=1}^\infty q^n \sigma_\alpha(n) = \sum_{n=1}^\infty \frac{n^\alpha q^n}{1-q^n}$

where $\alpha$ is any complex number and

$\sigma_\alpha(n) = (\textrm{Id}_\alpha*1)(n) = \sum_{d\mid n} d^\alpha \,$

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 $\mu(n)$:

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

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

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

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

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

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

Alternate form

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

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

where

$b_m = (a*1)(m) = \sum_{n\mid m} a_n\,$

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

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

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

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

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.