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

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.

• Relevant: math.stackexchange.com/q/703593 – Sal Feb 2 '15 at 9:10
• So your saying its absolutely possible? – remydib Feb 2 '15 at 19:10
• I don't understand your second sentence, but I think you should first ask: can you extend the Riemann zeta function to the quaternions? (This could be a good post.) I've never thought about it, but it's an interesting question. My guess is no--the Riemann zeta function is defined by a series in a half-plane, and extended to $\mathbb C$ by analytic continuation. You may still be able to make sense of the series for some region of quaternions, but I'm skeptical that it can be extended in a nice way to be a function of all quaternions. – Kimball May 12 '15 at 14:10
• The second sentence was basically: can we connect the dots by going through a higher dimension? – remydib Oct 21 '15 at 13:48
• @Kimball Every octonion has the form $re^{\theta{\bf u}}$ for some real $r\ge0$, angle $\theta$ and pure imaginary $\bf u$ of unit norm. Algebraically, $\bf u$ will function exactly like $i$ (it's a square root of negative one), so we can extend any holomorphic function defined on a domain in $\Bbb C$ to a domain in $\Bbb O$ by simply rotating around the real axis. – arctic tern Nov 2 '16 at 1:28