Fluctuations of the charged pion number in e+e- annihilation

It is shown how to find bounds on 〈n_ch^_^k(N)〉, the average of the k th power of the number of charged pions produced in the one-quantum annihilation e⁺e⁻->Nπ. The second moment, k=2, is studied in detail, also in its dependence on information on the first moment, k=1. The lower bound for k=2 appears to be of an experimentally significant magnitude. The I^G=0⁻ bounds are much tighter than those for 1⁺. It is further shown that the optimal bounds can be violated if I^G=1⁻ and especially if 0⁺-2⁺ interference plays a sizable role (as may happen in the two-photon process). A brief comment on overlinepp annihilation is also made, and an asymptotic (large N) formula is given for the upper bound of the neutral-to-charged pion ratio of a neutral system, for any pure isospin.