Let $(a_n)$ be a sequence of real numbers. The Lambert generating function of this sequence is defined as $$ L(x)=\sum_{n=1}^\infty a_n\frac{x^n}{1-x^n}. $$ I get why ordinary, exponential and Dirichlet generating functions are useful. But why are Lambert generating functions useful?
EDIT: rephrasing my question. What is an application of Lambert generating functions?
Lambert generating functions encode arithmetic information — they expose hidden recursive structures, bridge additive and multiplicative number theory, and sit at the intersection of series, products, and partitions.
A compelling application of Lambert generating functions is their ability to reveal surprising structure in arithmetic functions like the sum-of-divisors function, $\sigma_k(n)=\sum_{d|n}d^k$. For any positive integer $k$, the generating function identity $$ \sum_{n=1}^\infty \sigma_k(n)\,x^n = \sum_{n=1}^\infty \frac{n^k x^n}{1 - x^n} $$ expresses the ordinary generating function of $\sigma_k(n)$ as a Lambert series. This identity — derived by switching the order of summation in a double series — is cleanly explained here. The $k = 1$ case connects beautifully to Euler’s pentagonal number theorem, through its role in differentiating the generating function for the partition function, $\prod_{n=1}^\infty \frac{1}{1 - x^n}$. Taking a logarithmic derivative yields: $$ \frac{d}{dx} \log\left(\prod_{n=1}^\infty \frac{1}{1 - x^n}\right) = \sum_{n=1}^\infty \frac{n x^{n-1}}{1 - x^n}, $$ and multiplying by $x$ gives exactly the Lambert series for $\sigma_1(n)$. This leads to a striking recurrence involving generalized pentagonal numbers: $$ \sigma_1(n) = \sigma_1(n-1) + \sigma_1(n-2) - \sigma_1(n-5) - \sigma_1(n-7) + \cdots $$ with sign alternation and appropriate correction terms when $n$ is a pentagonal number. For a crisp explanation of this result and its connection to partitions, see Euler’s Pentagonal Number Theorem.
