 Lambert series

A Lambert series has the form ${\sum }_{n\ge 1}{a}_n{q}^n/(1-q^n)$. Expanding each denominator as a geometric series shows that its coefficient of $q^m$ is ${\sum }_{d\mid m}{a}_d$.

A Lambert series is a type of mathematical series named after the mathematician Johann Heinrich Lambert. It is defined in a particular form, usually involving a power series with specific coefficients. The general form of a Lambert series can be expressed as: \[ \sum_{n=1}^{\infty} \frac{n q^n}{1 - q^n} \] where \( |q| < 1 \) is a complex variable.