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, 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 $r\ge 1$, $d^{(r)}(N)\sim e^{-\gamma }(\log N{)}^{{\alpha }_r}$, as $N$ tends to infinity through superabundant numbers, where the exponents ${\alpha }_r$ are given by the Dickman--de Bruijn distribution. This yields explicit sets of divisors ${\mathcal{D}}_r(N)$ satisfying ${\sum }_{d\in {\mathcal{D}}_r(N)}d={\rho }_{r}N\log \log N+o(N)$ for each fixed $r$, where ${\rho }_r={\lim }_{n\to \infty }{A}_r(n)/n\log \log n$. We also compute exactly the contribution of the hook-shaped family of partitions $[r,1^l]$, denoted by ${\tilde{\rho }}_r$. Summed over all branches, these contributions account for approximately $91.85\%$ of $e^{\gamma }\approx 1.781072417990\dots$. The remaining proportion, coming from all other partitions, satisfies ${\lim }_{r\to \infty }{\rho }_r/({\rho }_r-{\tilde{\rho }}_r)=1.$