### 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

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

## 4 Comments:

Hi Kea,

Are any of these Michael E Hoffman Arxiv papers the same or an update of the 1997 paper?

Nine papers listed from 2000-present [two not dated].

Showing results 1 through 9 (of 9 total) for au:Hoffman_M

http://arxiv.org/find/math/1/

au:+Hoffman_M/0/1/0/all/0/1

Hi Doug. Those are more recent papers, but thanks for the link.

Great work by Carl (and Kea too). The Euler-Beta connecdtion is a happy discovery. I will be referring to Carl's post too.

Kea, I wouldn't call it a "derivation". At best that sort of thing would be a "postdiction".

I think that to appreciate what I'm doing with snuarks and masses and all that you really have to wait for another dozen posts or so. At best, all I'm doing now is assembling the tools and making vague arguments.

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