Vtyjkyuk

Has the $zeta(s)$ function, $sum_n 1/n^s$, been generalized
to quaternions, so $zeta(q)$ for $q$ a quaternion?

Euler defined it for $s$ integers, Chebyshev for $s$ real,
Riemann for $s$ complex. So it is natural to explore $s$ a quaternion. But perhaps this does not lead to an interesting Quaternion Hypothesis?

You can define $zeta(q)$ for any quaternion with real part $ne 1$ by plugging it into the Taylor series for $zeta(x)$ around a real point far enough from $1$ in the appropriate direction.

This is not very exciting, though, because every quaternion $q$ decomposes as $a+bell$ where $a$ and $b$ are real and $ell$ is a quaternion satisfying $ell^2=-1$. So for every fixed $q$, all terms in the Taylor series will lie in the complex plane spanned by $1$ and $ell$, and therefore no really new phenomena show up.

This shows that it doesn't matter which center for the Taylor series we choose, as long as it is closer to $q$ than it is to $1$. It also shows that we can extend the function continuously to quaternions with real part $1$ (other that $1$ itself).