Year: 2026
Type: preprint
Abstract: An equivalence of the Riemann Hypothesis due to Espinosa reveals a direct bridge to the theory of integer partitions. Building on this formulation, we analyze the asymptotic behavior of a family of combinatorial sums $A_r(n)$, called Espinosa's branches, and their realization by divisors of superabundant numbers. Our main result is an unconditional asymptotic formula for the associated cutoff divisors: for every fixed integer $r\geq 1$, $d^{(r)}(N)\sim e^{-γ}(\log N)^{α_r}$, as $N$ tends to infinity through superabundant numbers, where the exponents $α_r$ are given by the Dickman--de Bruijn distribution. This yields explicit sets of divisors $\mathcal{D}_r(N)$ satisfying $\sum_{d\in\mathcal{D}_r(N)}d=ρ_r N\log\log N+o(N)$ for each fixed $r$, where $ ρ_r = \lim_{n\to\infty} A_r(n)/n\log\log n $. We also compute exactly the contribution of the hook-shaped family of partitions $[r,1^l]$, denoted by $\tildeρ_r$. Summed over all branches, these contributions account for approximately $91.85\%$ of $e^γ\approx 1.781072417990\ldots$. The remaining proportion, coming from all other partitions, satisfies $ \lim_{r\to\infty} ρ_r/(ρ_r - \tildeρ_r) = 1. $
Source: arXiv (Cornell University)
Author: Segovia, Carlos
Institution: Universidad Nacional Autónoma de México
Language: English
Subfield: Algebra and Number Theory
Field: Mathematics
Domain: Physical Sciences
Open Access status: green
OpenAlex ID: W7127071580