Arithmetic Constraints and Limit Laws for Diagonal Rational Partitions
Abstract
Let RN(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 RN(⌊ρN⌋)=√(2ρ) N3/2−κ(ρ)N3/2/log N+o(N3/2/log N). The positive, continuously differentiable function κ is an explicit sum of lattice-distance integrals, with κ(1)≈ 0.00264713. For a uniform partition of n, all but oℙ(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≥ 4, but agrees with an independent model in every moment below order ⌈ N/2⌉. We identify the first discrepancy and prove a Gaussian limit. We also obtain joint denominator and size laws under two sampling rules.
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BibTeX
@misc{raghava2026arithmetic,
title = {{Arithmetic Constraints and Limit Laws for Diagonal Rational Partitions}},
author = {K. Srinivasa Raghava},
howpublished = {arXiv},
year = {2026},
month = sep,
doi = {10.48550/arxiv.2609.18945},
eprint = {2609.18945},
archivePrefix = {arXiv},
url = {https://arxiv.org/abs/2609.18945},
}
RIS
TY - UNPB
TI - Arithmetic Constraints and Limit Laws for Diagonal Rational Partitions
AU - K. Srinivasa Raghava
PY - 2026
DA - 2026/09/10
PB - arXiv
DO - 10.48550/arxiv.2609.18945
UR - https://arxiv.org/abs/2609.18945
AB - 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(⌊ρN⌋)=√(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)≈ 0.00264713. For a uniform partition of n, all but o_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≥ 4, but agrees with an independent model in every moment below order ⌈ N/2⌉. We identify the first discrepancy and prove a Gaussian limit. We also obtain joint denominator and size laws under two sampling rules.
ER -