The Riemann Hypothesis through the looking of partitions

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, expressed in terms of monomial symmetric polynomials. In this work, we propose that the Riemann Hypothesis splits into the successive realization of each Espinosa branch by a concrete subset of divisors. In this direction, we rigorously establish the first step: the branch $r=1$ is fully realized. For the higher branches, we define the asymptotic proportions ${\rho }_r={\lim }_{n\to \infty }{A}_r(n)/(n\log \log n)$ and we compute exactly the contribution of the hook-shaped family of partitions $[r,1^l]$, denoted by ${\tilde{\rho }}_r$, which accounts for $91.85\%$ of the total classical constant $e^{\gamma }\approx 1.781072417990\dots$ The remaining proportion, coming from all other partitions, satisfies ${\lim }_{r\to \infty }{\rho }_r/({\rho }_r-{\tilde{\rho }}_r)=1$. Assuming the Alaoglu-Erdös conjecture, we use the list of 10,000 existing colossally abundant numbers (OEIS A004490, A073751) to yield explicit cutoff divisors realizing the first seven Espinosa branches. We present computational evidence showing how these first realizations appear, supporting the proposed divisor-realization framework.