Has the Riemann Hypothesis been generalized to the Octonions and the Quaternions?

Question

I've noticed that it uses imaginary numbers. I know that sometimes when I have too few dimensions like (-1)^n, dots show where I might expect lines due to imaginary numbers. So perhaps there is a function lost to the other normed divisions? I know there is the upper limit of the octonions for normed divisons, so I was wondering if people have explored that aspect, and it would allow us to quickly rule it out.

Answer 1

Any holomorphic function which is analytic and real valued on $\mathbb{R}$ minus some discrete set of poles has a natural extension to quaternions by plugging quaternions into the Taylor series around any real value in its domain. These functions are symmetric across the real axis, i.e. $f(\overline{z})=\overline{f(z)}$. But extending to quaternions does not provide anything interesting: anytime $f(x+yi)=u+vi$, we have $f(x+y\mathbf{t})=u+v\mathbf{t}$ for any unit imaginary quaternion $\mathbf{t}$. (The unit imaginary quaternions are precisely the square roots of $-1$, so they behave algebraically and analytically the same as $i$. Indeed, the numbers of the form $x+y\mathbf{t}$ form an isomorphic copy of $\mathbb{C}$ in $\mathbb{H}$.) So the quaternion zeros of the Riemann zeta function are nothing more special than the complex zeros.