Arithmetic Constraints and Limit Laws for Diagonal Rational Partitions

Abstract

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 positive, continuously differentiable function $κ$ is an explicit sum of lattice-distance integrals, with $κ(1)\approx 0.00264713$. For a uniform partition of $n$, all but $o_{\mathbb{P}}(n/\log n)$ prime-denominator blocks in $(n/2,n]$ have total 2 or 3, according to whether $p/n$ lies below or above an explicit threshold near 0.7522. For fixed $N$, we give the Ehrhart numerator and determine how its poles control quasipolynomial coefficients. Its residue distribution is asymmetric for $N\ge 4$, but agrees with an independent model in every moment below order $\lceil N/2\rceil$. We identify the first discrepancy and prove a Gaussian limit. We also obtain joint denominator and size laws under two sampling rules.