We study the fluctuation of the number of particles in ideal Bose-Einstein
condensates, both within the canonical and the microcanonical ensemble. Emp
loying the Mellin-Barnes transformation, we derive simple expressions that
link the canonical number of condensate particles, its fluctuation, and the
difference between canonical and microcanonical fluctuations to the poles
of a Zeta function that is determined by the excited single-particle levels
of the trapping potential. For the particular examples of one- and three-d
imensional harmonic traps we explore the microcanonical statistics in detai
l, with the help of the saddle-point method. Emphasizing the close connecti
on between the partition theory of integer numbers and the statistical mech
anics of ideal Bosons in one-dimensional harmonic traps, and utilizing ther
modynamical arguments, we also derive an accurate formula for the fluctuati
on of the number of summands that occur when a large integer is partitioned
. (C) 1998 Academic Press.