M Theory Lesson 143

Putting together the Hoffman and Castro expressions, for real $s$ and $t$ in the critical interval with $|s+t| < 1$, we obtain

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

where the left hand side is the expression

$\sum_{m} \frac{s^m}{m!} \sum_{k_1,k_2,\cdots,k_m} \frac{1}{k_1 k_2 \cdots k_m} \sum_{n} \frac{t^n}{(k_1 + k_2 + \cdots + k_m)^{n}}$

Specific values of the zeta function include, for the choice $t = 0.5$, $\zeta (1.5) = 2.612$ and, using the functional equation,

$\zeta (- \frac{1}{2}) = \frac{1}{\sqrt{2}} \pi^{\frac{-3}{2}} \Gamma (\frac{3}{2}) \zeta (\frac{3}{2})$

so that the centre ratio in the first equation above becomes

$\sqrt{2} \pi^{\frac{3}{2}} \frac{2}{\sqrt{\pi}} = 2 \sqrt{2} \pi$

giving a particularly interesting relation for the parameter $s < \frac{1}{2}$ involving the expression

$\frac{\zeta (0.5 - s)}{\zeta (0.5 + s)} \frac{\zeta (s)}{\zeta (1-s)}$

It would be nice to extend this to complex values of the parameters, because zeroes of the zeta function occur in conjugate pairs and the finite positivity of an MZV could then rule out zeroes lying in this region.