Lambert series

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For generalized Lambert series, see Appell–Lerch sum.
Function , 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

It can be resummed formally by expanding the denominator:

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

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


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

where is the number of positive divisors of the number n.

For the higher order sigma functions, one has

where is any complex number and

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 :

For Euler's totient function :

For Liouville's function :

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

Alternate form[edit]

Substituting one obtains another common form for the series, as


as before. Examples of Lambert series in this form, with , occur in expressions for the Riemann zeta function for odd integer values; see Zeta constants for details.

Current usage[edit]

In the literature we find Lambert series applied to a wide variety of sums. For example, since is a polylogarithm function, we may refer to any sum of the form

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

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[edit]