Lambert series

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 a_n 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.