The logarithmic distribution of leading digits and finitely additive measures

When the ‘1’ key on my old computer gave out I was not surprised. That this particular key was first to break was just another manifestation of the long observed phenomenon, namely that more numbers begin with digit 1 than with any other digit. The empirical logarithmic distribution law states that for a ‘randomly chosen’ number, the leading digit will be 1 with probability log,,,2. In general, the leading digit d occurs with probability log,,,(l + l/d), and in fact for any positive integer k, the probability that the decimal expansion of a number begins with k is log,,,(l + l/k). This empirical logarithmic law was first observed and formulated by Simon Newcomb in 1881 [8]: ‘That the ten digits do not occur with equal frequency must be evident to any one making much use of logarithmic tables, and noticing how much faster the first pages wear out than the last ones.’ ‘The law of probability of the occurrence of numbers is such that all mantissae of their logarithms are equally probable.’ A number of papers have been written to deal with this ‘first digit phenomenon’, giving explanations using various summation methods or definitions of probability. On close inspection, all these methods attempt to introduce a (finitely additive) probability measure for which the distribution of leading digits satisfies the logarithmic law. In this paper we state necessary and sufficient conditions for a probability measure to satisfy the first digit law, and discuss various results and proposed explanations in the light of these conditions.