Transcendence II

In the last post Carl Brannen (who has derived the number of particle generations in his latest blog post) said he had once played with the value $\zeta (3)$ and found that it could not be a simple rational multiple of $\pi^3$. In his classic paper [1] on MZVs, Hoffman mentions a number of conjectures based on a consideration of the zeta function as a map into the reals from a subalgebra of the noncommutative polynomial ring $\mathbb{Q}[x,y]$ on two letters (with both ordinary product and a shuffle type product), namely the subalgebra of polynomials of the form $\mathbb{Q}.1 + xTy$ for any $T$.

The conjecture states that the quotient of this algebra by the kernel of $\zeta$ is a simple polynomial algebra on some set of Lyndon words. If true, it would imply in particular that $\zeta (3)$ cannot be a rational multiple of $\pi^3$. Hoffman shows that $\zeta (x^{m} y^{n})$ can always be written as a simple combination of Riemann zeta values $\zeta (i)$ for $i \geq 2$. The theorem amounts to showing that, for real numbers $s$ and $t$ lying in the basic simplex (bounded by the line $s + t = 1$)

$\sum_{m,n} s^m t^n \zeta (x^{m} y^{n}) = 1 - \frac{\Gamma (1 - s) \Gamma (1 - t)}{\Gamma (1 - s - t)}$

where M theorists will recognise the Euler beta function that appears in the Veneziano amplitude.

[1] M. E. Hoffman, J. Alg. 194 (1997) 477-495