Lambert Series

Manuel Eberl 📧

November 24, 2023

This is a development version of this entry. It might change over time and is not stable. Please refer to release versions for citations.

Abstract

This entry provides a formalisation of Lambert series, i.e. series of the form $L(a_n, q) = \sum_{n=1}^\infty a_n q^n / (1-q^n)$ where $a_n$ is a sequence of real or complex numbers.

Proofs for all the basic properties are provided, such as:

  • the precise region in which $L(a_n, q)$ converges
  • the functional equation $L(a_n, \frac{1}{q}) = -(\sum_{n=1}^\infty a_n) - L(a_n, q)$
  • the power series expansion of $L(a_n, q)$ at $q = 0$
  • the connection $L(a_n, q) = \sum_{k=1}^\infty f(q^k)$ for $f(z) = \sum_{n=1}^\infty a_n z^n$
  • that links a Lambert series to its ``corresponding'' power series
  • connections to various number-theoretic functions, e.g. the divisor $\sigma$ function via $\sum_{n=1}^\infty \sigma_{\alpha}(n) q^n = L(n^\alpha, q)$

The formalisation mainly follows the chapter on Lambert series in Konrad Knopp's classic textbook Theory and Application of Infinite Series and includes all results presented therein.

License

BSD License

Topics

Session Lambert_Series