Conjecture: $$\left[n\in\mathbb{Z}^+,z\in\mathbb{C},0=\sum_{k=0}^\infty \frac{z^k}{\left(nk\right)!}\right]\Rightarrow z\in\mathbb{R}$$ This conjecture has been verified for $n\in\{1,2,4\}$.
The motivation for this conjecture arose during the study of the exponential sum function which has applications to exponentiation in rings with abelian multiplication: $$\text{rues}_n\left(z\right)=\sum_{k=0}^\infty \frac{z^{nk}}{\left(nk\right)!}=\frac{1}{n}\sum _{k=1}^n \exp\left(ze^{2ki\pi/n}\right)$$
It has been shown that the zeros of the Mittag-Leffler function $$E_{\alpha}(z)\stackrel{\text{def}}{=}\sum_{n=0}^{\infty}\frac{z^n}{\Gamma(\alpha n+1)}\quad(\alpha > 0)$$ are real and negative whenever $\alpha\geq 2$.
See Wiman (1905) and Pólya (1921).
Wiman, A. 1905. “Über die Nullstellen der Funktionen $E_a(x)$.” Acta Mathematica 29: 217–34.
Pólya, G. 1921. “Bemerkung Über Die Mittag-Lefflerschen Funktionen $E_a(z)$.” Tohoku Mathematical Journal, First Series 19: 241–48.
