On cellular rational approximations to $\zeta(5)$

TL;DR

This paper studies a family of integrals related to moduli spaces that produce rational approximations to zeta(3) and zeta(5), achieving a sequence with a specific approximation quality for zeta(5).

Contribution

It introduces a new method using cellular integrals and symmetry groups to generate effective rational approximations to ta(5).

Findings

Constructed an infinite sequence of rational approximations to ta(5).

Achieved an approximation bound of |ta(5)-p/q| < 1/q^{0.86}.

Linked integrals on moduli space to zeta value approximations.

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 $\zeta(3)$ and $\zeta(5)$. By exploiting the action of a large symmetry group on these integrals, we construct an infinite $effective$ sequence of rational approximations $p/q$ to $\zeta(5)$ satisfying \[ 0<\bigg|\zeta(5)-\frac pq\bigg|<\frac1{q^{0.86}}. \]