Extension to prime codes below higher ordinal powers

Extend the asymptotic counting method to prime codes of ordinals below \(\omega^{\omega^k}\) for every fixed \(k\ge 3\), including the higher-dimensional lattice zeta functions, the higher-order poles introduced by the prime weights, and the longer expansions and estimates required for the analysis.

Background

The paper proves a strong asymptotic formula for integers whose prime factors are indexed by $r^a s^b$, and identifies the case $(r,s)=(2,3)$ with the prime codes of ordinals below $\omega^{\omega^2}$. The authors then describe the analogous counting problem for prime codes below $\omega^{\omega^k}$ when $k\ge3$.

For these higher-dimensional problems, the relevant lattice zeta function becomes $k$-dimensional and the prime-weight perturbations produce higher-order poles in the Mellin kernel. The paper indicates that the general method is expected to extend, but explicitly leaves the necessary expansions and estimates unresolved.