Arithmetic Dijkgraaf-Witten invariants for real quadratic fields, quadratic residue graphs, and density formulas

We compute Hirano’s formula for the mod 2 arithmetic Dijkgraaf-Witten invariant $Z_k$ for the ring of integers of the quadratic field $k=\mathbb{Q}(\sqrt{p_1\cdots p_r})$, where $p_i$'s are distinct prime numbers with $p_i\equiv 1(\mathrm{m}\mathrm{o}\mathrm{d}4)$, and give a simple formula for $Z_k$ in terms of the graph obtained from quadratic residues among $p_1,\cdots ,p_r$. Our result answers the question posed by Ken Ono. We also give a density formula for mod 2 arithmetic Dijkgraaf-Witten invariants.