Lambert series

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_{d\mid m} a_d\,

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.

See also

References