Lambert series

Summary

A Lambert series is a series of the form $\sum_{n \ge 1} a_n \frac{q^n}{1 - q^n}$ in a variable $q$ with $\abs{q} < 1$. Expanding each fraction as a geometric series turns it into a power series whose coefficients are divisor sums $\sum_{d \mid m} a_d$. This makes Lambert series a bridge between arithmetic functions and power series, used in the theory of partitions, modular forms and sums of squares.