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 diffe rent origins shows a marked asymmetry in favor of small digits that goes un der the name of Benford's law. We analyze in detail this property for diffe rent data sets and give a general explanation for the origin of the Benford 's law in terms of multiplicative processes. We show that this law can be a lso generalized to series of numbers generated from more complex systems li ke the catalogs of seismic activity. Finally, we derive a relation between the generalized Benford's law and the popular Zipf's law which characterize the rank order statistics and has been extensively applied to many problem s ranging from city population to linguistics. (C) 2001 Published by Elsevi er Science B.V.