Nonasymptotic Properties of Roots of a Mittag-Leffler Type Function

We completely solve the problem of finding the number of positive and nonnegative roots of the Mittag-Leffler type function
$$ E_\rho(z;\mu)=\sum_{n=0}^\infty \frac{z^n}{\Gamma(\mu+n/\rho)}, \qquad \rho>0, \qquad \mu\in\mathbb C, $$
for $\rho>1$ and $\mu\in\mathbb R$. We prove that there are no roots in the left angular sector $\pi/\rho\le|\arg z|\le\pi$ for $\rho>1$ and $1\le\mu<1+1/\rho$. We consider the problem of multiple roots; in particular, we show that the classical Mittag-Leffler function $E_n(z;1)$ of integer order does not have multiple roots.