Title: Benford’s Law: The Significant-digit Phenonemon

Abstract:

A century-old empirical observation called Benford´s Law states that the significant digits of many datasets are logarithmically distributed, rather than uniformly distributed as might be expected. For example, more than 30% of the leading decimal digits are 1, and fewer than 5% are 9. This talk will briefly survey some of the colorful history and empirical evidence of Benford´s law, and then discuss recent mathematical discoveries that help explain the ubiquity of Benford datasets. For example, it has now been shown that iterations of many common functions (including polynomials, power, exponential, and trigonometric functions, and compositions thereof), geometric Brownian motion (hence many stock market models), large classes of ODE’s, random mixtures of data from different sources, finite-state Markov chains, and many numerical algorithms like Newton´s method, all produce Benford distributions. Applications of these theoretical results to practical problems of fraud detection, analysis of round-off errors in scientific computations, and diagnostic tests for mathematical models will be mentioned, as well as several open problems in dynamical systems, probability, number theory, and differential equations.