Proof of the Borwein-Broadhurst conjecture for a dilogarithmic integral arising in quantum field theory

TL;DR

This paper provides a formal proof of a conjecture linking a dilogarithmic integral in hyperbolic geometry to a special value of a Dirichlet L series, confirming numerical evidence with advanced mathematical techniques.

Contribution

The paper offers the first rigorous proof of the Borwein-Broadhurst conjecture connecting hyperbolic volume integrals to Dirichlet L series values, building on Zagier's formulas.

Findings

Confirmed the conjecture with a formal proof.

Validated the integral-L series relationship numerically to high precision.

Connected hyperbolic geometry with special values of L series.

Abstract

Borwein and Broadhurst, using experimental-mathematics techniques, in 1998 identified numerous hyperbolic 3-manifolds whose volumes are rationally related to values of various Dirichlet L series $\textup{L}_{d}(s)$. In particular, in the simplest case of an ideal tetrahedron in hyperbolic space, they conjectured that a dilogarithmic integral representing the volume equals to $\textup{L}_{-7}(2)$. Here we have provided a formal proof of this conjecture which has been recently numerically verified (to at least 19,995 digits, using 45 minutes on 1024 processors) in cutting-edge computing experiments. The proof essentially relies on the results of Zagier on the formula for the value of Dedekind zeta function $\zeta_{\mathbb{K}}(2)$ for an arbitrary field $\mathbb{K}$.