Arithmetic Constraints and Limit Laws for Diagonal Rational Partitions

Let $R_N(m)$ count unordered partitions of $m$ into reduced positive fractions whose numerators and denominators are at most $N$, excluding integer parts. Uniformly for $ρ$ in compact positive intervals, $\log R_N(\lfloorρN\rfloor)=\sqrt{2ρ}\,N^{3/2}-κ(ρ)N^{3/2}/\log N+o(N^{3/2}/\log N)$. The positi