  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.

Soft Condensed Matter