# Conjecture: all complex roots of $\sum_{k=0}^\infty \frac{z^k}{\left(nk\right)!}$ are real

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: $$\text{rues}_n\left(z\right)=\sum_{k=0}^\infty \frac{z^{nk}}{\left(nk\right)!}=\frac{1}{n}\sum _{k=1}^n \exp\left(ze^{2ki\pi/n}\right)$$

• $e^z$ does not have any roots and thus $n=1$ is vacuously true. $0\neq e^{2ki\pi}=1$. May 11 '19 at 0:26

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 $$\alpha\geq 2$$.
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.