2 7 A ug 1 99 8 The Uneven Distribution of Numbers in Nature

Suppose you look at today's stock prices and bet on the value of the first digit. One could guess that a fair bet should correspond to the frequency of 1/9 = 11.11% for each digit from 1 to 9. This is by no means the case because one can easily observe a strong prevalence of the small values over the large ones. The first three integers 1,2 and 3 alone have globally a frequency of 60% while the other six values 4, 5, 6, 7, 8 and 9 appear only in 40% of the cases. This situation is actually much more general than the stock market and it occurs in a variety of number catalogs related to natural phenomena. The first observation of this property traces back to G. Benford in 1938. He investigated 20 tables of numbers ranging from the area of lakes and the length of rivers to the molecular weights of molecular compounds. In all cases he found the same behavior for which he guessed the probability distribution P (n) = log[(n + 1)/n] where n is the value of the first integer. Since then this observation has remained marginal and seldomly reported as a mathematical curiosity. In this note we revamp these sparse observations with a specific application to the stock market. We also argue that the origin of this uneven distribution can be found in the multiplicative nature of fluctuations in economics and in many natural phenomena. This brings us close to the problem of the spontaneous origin of scale invariant properties in various phenomena which is a debated question at the frontier of different fields. Consider the values of the Madrid, Vienna and Zurich stock markets of January 23, 1998. The stock prices N are expressed in the local currencies. If the values of N would be randomly distributed we would expect uniform distribution for the value of the first digit n = 1,