Lambert series
In mathematics, a Lambert series, named for Johann Heinrich Lambert, is a series taking the form S ( q ) = ∑ n = 1 ∞ a n q n 1 − q n . {\displaystyle 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 ) = ∑ n = 1 ∞ a n ∑ k = 1 ∞ q n k = ∑ m = 1 ∞ b m q m {\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 an with the constant function 1(n) = 1: b m = ( a ∗ 1 ) ( m ) = ∑ n ∣ m a n .