Let $a_n \in \Bbb{R}, n \in \Bbb{Z}$ be a bi-infinite sequence. For any set $T\subset \Bbb{Z}$, define a divisibility sieve $\widehat{T}$ to be a transformation of the sequence that deletes every element $a_{tm}$ for all $t \in T$, $m \in \Bbb{Z}$. Define the characteristic function of $\widehat{T}$ to be $\chi_T(n) = 1$ if $n \in T\Bbb{Z} = \cup_{t \in T} t\Bbb{Z}$ and $0$ otherwise.
Then $\chi_T(n)$ is periodic if and only if $T$ is contained in a finite union of prime ideals of $\Bbb{Z}$.
Any idea how to prove this? Seems true.
The period will be the lowest common multiple of the generators of the ideals making up the union.
However, for the converse, if $T=\{5\}$ then $\chi_T$ is periodic, but in what sense is $T$ a union of ideals? Am I missing something obvious?