The equation $\zeta(q)=0$ for $q$ a quaternion

Question

I know there have been several attempts to define a theory of functions of a quaternionic variable. I would like to know if a coherent and satisfying definition of the "Riemann" zeta function exists in this framework and if studying the equation $\zeta(q)=0$ for $q$ a quaternion could lead to some progress towards RH. If I'm not mistaken the principle of isolated zeros doesn't hold true anymore, but do we conjecture that the solutions of the above equation lie in a plane?

Answer 1

For $M \in M_k(\Bbb{C})$ and $\|\exp(-M)\|< 1$ and $\zeta(M) = \sum_{n=1}^\infty \exp(-M \ln n)$ and by analytic continuation for every $M$

• If $M=PD P^{-1}$ is diagonalizable then $$\zeta(M) = P \zeta(D)P^{-1},\qquad \zeta(D)= \pmatrix{\zeta(D_{11})& &\\ & \zeta(D_{22})& \\ & & \ddots} $$

• If $M$ is not diagonalizable then the Jordan normal form gives $M = P (D+N) P^{-1}$ where $ND=DN, N^k=0$ and $$\zeta(M) = P \sum_{m=0}^{k-1} \frac{N^m}{m!} \zeta^{(m)}(D) P^{-1}$$

• The quaternions are isomorphic to a diagonalizable subalgebra of $M_2(\Bbb{C})$