Sums of the Powers of the Zeros of the Mittag-Leffler Function

The Mittag-Leffler function, which is a generalization of the exponential function, occurs naturally in the solution of dynamic problems involving fractional calculus. For example see the book Fractional Differential Equations by Podlubny, or for a specific application see the problem of the fractional oscillator by Achar et. al. Physica A297 (2001) 361 and A309 (2002) 275. The zeros of the Mittag-Leffler function play a significant role in the dynamic solutions. However, very little is known about these zeros. A summary of the available information about the zeros of Mittag-Leffler functions will be given and new results pertaining to the sums of the powers of the zeros will be presented.