Sums of the Zeros of the Single Parameter Mittag-Leffler Function With the Parameter Equal to One-Half

The sums of the powers of the reciprocals of the non-trivial (non-zero) zeros of the single parameter Mittag-Leffler function with alpha equal to one-half have been investigated. Analytical results for these sums were compared to the results from numerically summing over a billion zeros. Sums for integer powers from 3 to 10 agreed with the predictions of the analytical results but not for powers 1 and 2. The sum of the reciprocals of the zeros diverged in contrast to the analytical result and the sum of the squares of the zeros converged to a different result than predicted. This illustrates that the analytical results for polynomials cannot in general be applied to the infinite series representation of the Mittag-Leffler functions.