  We study the occupation numbers and number fluctuations of ultra-cold atoms
in deep optical lattices for finite temperatures within the Bose-Hubbard model.
Simple analytical expressions for the mean occupation number and number
fluctuations are obtained in the weak-hopping regime using an interpolation
between results from different perturbation approaches in the Mott-insulator
and superfluid phases. These analytical results are compared to exact one
dimensional numerical calculations using a finite temperature variant of the
Density-Matrix Renormalisation Group (DMRG) method and found to have a high
degree of accuracy. We also find very good agreement in the crossover
``thermal'' region. With the present approach the magnitude of number
fluctuations under realistic experimental conditions can be estimated and the
properties of the finite temperature phase diagram can be studied.
