Asymptotic order of the expected Frobenius number in the ER-type model

Determine the asymptotic order of the expected Frobenius number E[F((p))] for the unconstrained Erdős–Rényi-type random numerical semigroup model (p), specifically by proving that E[F((p))] = Θ((1/p) log(1/p)).

Background

After improving upper bounds to within polylogarithmic factors of previously known lower bounds, the authors’ experiments suggest a sharper asymptotic order for the expected Frobenius number in the unconstrained ER-type model.

They explicitly conjecture that the true growth is linear in 1/p times a single logarithmic factor, contrasting with earlier quadratic or cubic log factors in upper bounds.

References

However, extensive experiments , in which 1000 Erd\H{o}s-R enyi semigroups were generated for each of fifteen values of $p$, suggest the following conjecture.\n\nConjecture.\n\n1. The expected Frobenius number $\mathbb{E}[\F((p))]$ is of order $\frac{1}{p} \log \left(\frac{1}{p}\right)$.

— Improved Upper Bounds on Key Invariants of Erdős-Rényi Numerical Semigroups (2411.13767 - Bogart et al., 21 Nov 2024) in Section 5 (Experiments, Conclusions, and Future Work)