Linear recurrences and rational Lambert series

Abstract

For a sequence ${\hat{I}}^3=({\hat{I}}_n^3{)}_{n\ge 1}$, define \[ L_γ(z):=\sum_{n\ge 1}γ_n\frac{z^n}{1-z^n} =\sum_{n\ge 1}\Bigl(\sum_{d\mid n}γ_d\Bigr)z^n. \] We prove a short rigidity theorem: if ${\hat{I}}^3$ is eventually linearly recurrent and $L_{\hat{I}}^3(z)$ is rational, then ${\hat{I}}^3$ is finitely supported. Equivalently, among sequences with rational ordinary generating function, the only ones whose Lambert series is rational are the finitely supported sequences. The proof specializes the data at a finite place of a finitely generated ring and then uses the periodicity of recurrences over finite fields.