Explaining the uneven distribution of numbers in nature: the laws of Benford and Zipf

The distribution of first digits in numbers series obtained from very different origins shows a marked asymmetry in favor of small digits that goes under the name of Benfordʹs law. We analyze in detail this property for different data sets and give a general explanation for the origin of the Benfordʹs law in terms of multiplicative processes. We show that this law can be also generalized to series of numbers generated from more complex systems like the catalogs of seismic activity. Finally, we derive a relation between the generalized Benfordʹs law and the popular Zipfʹs law which characterize the rank order statistics and has been extensively applied to many problems ranging from city population to linguistics.