Explanation for the appearance of the spectral zeta function

Explain why the number-theoretic spectral zeta function of the integer lattice appears in the equation determining the longest-increasing-subsequence exponent α(p).

Background

The exponent α(p) is identified as the unique solution of an equation that can be expressed using the spectral zeta function of the integer lattice. The paper derives this formula but does not provide a conceptual explanation for why this particular number-theoretic function governs the exponent.

This unresolved issue concerns the structural origin of the exponent formula rather than its proof or numerical evaluation.

References

However, at present, we have no explanation for the appearance of the number-theoretic function $\zeta_{\mathbb{Z}$.

— The longest increasing subsequence of Brownian separable permutons  (2506.19123 - Adhikari et al., 23 Jun 2025) in Section 1.1, paragraph following the product representation of the exponent equation