On cellular rational approximations to $ζ(5)$

TL;DR

This work develops a geometric and arithmetic framework of cellular integrals on $\mathcal{M}_{0,8}$ to produce simultaneous rational approximations to $\zeta(5)$ and $\zeta(3)$. By leveraging a large Rhin–Viola–type symmetry group and a duality with well-poised hypergeometric series, the authors extract an explicit infinite family of effective rational approximations to $\zeta(5)$ with an exponent $0.86$, and they demonstrate a concrete instance achieving $\gamma\approx0.86597$. Central to the method are motivic interpretations of the integrals, Barnes-type representations, and an intricate group action on the parameter spaces that governs denominator growth and the asymptotics. The results illuminate a rich interplay between geometry, hypergeometric transformations, and the arithmetic of zeta-values, suggesting a general theory for cellular integrals with potential applications to higher weights and irrationality questions. The paper also establishes structural dualities and symmetry saturations that strongly constrain the asymptotics and arithmetic of the associated linear forms.

Abstract

We analyse a certain family of cellular integrals, which are period integrals on the moduli space $\mathcal{M}_{0,8}$ of curves of genus zero with eight marked points, and give rise to simultaneous rational approximations to $ζ(3)$ and $ζ(5)$. By exploiting the action of a large symmetry group on these integrals, we construct an infinite $effective$ sequence of rational approximations $p/q$ to $ζ(5)$ satisfying \[ 0<\bigg|ζ(5)-\frac pq\bigg|<\frac1{q^{0.86}}. \]

Figures (2)

• Figure 1: The graph $G$.
• Figure 2: The complete graph $K_4$ on 4 vertices labelled $s_0,\ldots, s_3$ is depicted on the left as a tetrahedron. The two front faces have been subdivided into triangles. Subdividing all faces leads to the graph $G_4$ with 6 vertices and 12 edges shown on the right.

Theorems & Definitions (1)

• Theorem 1