Lambert summation

Summary

Lambert summation is a summability method built on Lambert series $\sum n a_n \frac{x^n}{1 - x^n}$, multiplied by $1 - x$, with $x \to 1$ from below. It is regular, so it gives the usual sum to every convergent series. It is closely tied to the prime number theorem. The series $\sum_{n \ge 1} \mu(n)/n$ is Lambert summable to $0$ by a short computation, and a Tauberian theorem upgrades this to convergence.