  Zipf's law describes the empirical size distribution of the components of
many systems in natural and social sciences and humanities. We show, by solving
a statistical model, that Zipf's law co-occurs with the maximization of the
diversity of the component sizes. The law ruling the increase of such diversity
with the total dimension of the system is derived and its relation with Heaps'
law is discussed. As an example, we show that our analytical results compare
very well with linguistics datasets.
