I'm checking over some of my work from studying prime numbers, and I found where I used the Hurwitz Zeta function $$\zeta(s,\alpha)_{\alpha \in [0,1)} = \sum_{n\in \mathbb{N}} \frac{1}{(n+\alpha)^s}, $$ to represent the sum of the inverses of integers in a specific residue class
If we let $\alpha \in [0,1) \cap {\mathbb{Q}} $, then $\exists a < q : \alpha := \frac{a}{q}$ and if we let $T$ be the residue class $\{ a \pmod{q}\}$ we could form the sum $$ \sum_{n\in T} \frac{1}{n^s} = \sum_{n \in \mathbb{N}}{\frac{1}{(qn+a)^s}} = \frac{1}{q^s}\sum_{n\in \mathbb{N}}{\frac{1}{(n + \alpha)^s}} = \frac{1}{q^s} \zeta(s,\alpha). $$
Is the sum irrational in general for all residue classes $T$, when $Re(s) > 1$?