Linear recurrences and rational Lambert series

TL;DR

The paper proves that for sequences with rational Lambert series, being eventually linearly recurrent implies the sequence must be finitely supported, highlighting a rigidity property.

Contribution

It establishes a new rigidity theorem linking rational Lambert series and linear recurrences, showing only finitely supported sequences have this property.

Findings

Sequences with rational Lambert series and linear recurrence are finitely supported.

The proof uses specialization at finite places and periodicity over finite fields.

Rational generating functions imply finiteness for sequences with rational Lambert series.

Abstract

For a sequence $\gamma=(\gamma_n)_{n\ge 1}$, define \[ L_\gamma(z):=\sum_{n\ge 1}\gamma_n\frac{z^n}{1-z^n} =\sum_{n\ge 1}\Bigl(\sum_{d\mid n}\gamma_d\Bigr)z^n. \] We prove a short rigidity theorem: if $\gamma$ is eventually linearly recurrent and $L_\gamma(z)$ is rational, then $\gamma$ 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.