Tests of conjectures on multiple Watson values

TL;DR

This paper introduces multiple Watson values (MWVs) as specific iterated integrals related to special algebraic numbers and conjectures their dimensional structure, supported by extensive numerical relation searches up to weight 5.

Contribution

The paper defines MWVs and proposes a conjecture on their dimensionality, backed by high-precision numerical evidence and relation searches, advancing understanding of their algebraic structure.

Findings

Conjecture on the dimension of MWV space matches numerical data

Performed 6639 integer relation searches at high precision

Supported the conjecture up to weight 5

Abstract

I define multiple Watson values (MWVs) as iterated integrals, on the interval $x\in[0,1]$, of the 6 differential forms $A=d\log(x)$, $B=-d\log(1-x)$, $T=-d\log(1-z_1x)$, $U=-d\log(1-z_2x)$, $V=-d\log(1-z_3x)$ and $W=-d\log(1-z_4x)$, where $z_1=\gamma^2$, $z_2=\gamma/(1+\gamma)$, $z_3=\gamma^2/(1-\gamma)$ and $z_4=\gamma=2\sin(\pi/14)$ solves the cubic $(1-\gamma^2)(1-\gamma)=\gamma$. Following a suggestion by Pierre Deligne, I conjecture that the dimension of the space of ${\mathbb Z}$-linearly independent MWVs of weight $w$ is the number $D_w$ generated by $1/(1-2x-x^2-x^3)=1+\sum_{w>0}D_w x^w$. This agrees with 6639 integer relation searches, of dimensions up to $D_5+1=85$, performed at 2000-digit precision, for $w<6$.