Wilson's theorem states $n \in \mathbb N$ is prime iff $(n-1)! \equiv -1\pmod n$. The $\Gamma$-function extends the usual factorial to complex numbers.
What are the complex numbers such that $\Gamma(z)+1 = nz$ , $n \in \mathbb Z$?
Eisenstein or Gaussian primes don't necessarily satisfy the requirement, take for example $2+\omega$ and $5+12i$ respectively.
What I've tried:
Let $z=a+ib$. From the definition of the $\Gamma$-function, we have
$\Gamma(z)=(z-1)\Gamma(z-1)$.
$\Gamma(a+ib)=(a-1+ib)\Gamma(a-1+ib)$
$=(a-1+ib)(a-2+ib)\Gamma(a-2+ib)$
$=(a-1+ib)(a-2+ib)\cdots(a-k+ib)\Gamma(a-k+ib)$
$=\Gamma(ib)\prod_{k=0}^{a-1}{(k+ib)}$
Now, turning to the imaginary-$\Gamma$, a brick wall I ran into...
$$\Gamma(ib)= \int_0^{\infty}\frac{t^{-1+ib}}{e^t}\mathrm{d}t$$
... and cannot evaluate.
Questions
- How do we evaluate $\Gamma(a+ib)$?
- How should we go about solving for $z$ once 1. is done?
Computational 'evidence'
Wolfram|Alpha thinks these $z$ exist, infact they seem plentiful. I'm not sure if approximation is muddling the results, but I doubt it.
I'm using solve Gamma(a+ib) + 1= n(a+ib) and plugging in values of $a,b,n$.